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DYNAMIC FRACTURE PROPERTIES OF CEMENTITIOUS MATERIALS 



By 
DAVID EDWARD LAMBERT 



A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL 

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT 

OF THE REQUIREMENTS FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 

UNIVERSITY OF FLORIDA 
1998 



This dissertation and research effort is dedicated to my wife, Deborah, for her 
patience and understanding throughout my educational pursuits as well as her 
encouragement to do so. Special dedication is made to my brother, John, and brother-in- 
law, Jimmy, whose deaths have taught me that, while success in life might be measured by 
milestones and goals, it is defined and given meaning through personal happiness and 
enjoyment of family. 



ACKNOWLEDGMENTS 

The author would like to acknowledge the advice and counsel given to him by his 
faculty supervisory committee: Professors E.K. Walsh, D.M. Belk, J.E. Milton, and C.S. 
Anderson. A most special acknowledgment is extended to Professor C. A. Ross, 
supervisory committee chair, whose encouragement, mentorship, advice, and friendship 
made this research possible. 

Acknowledgment is made to Mr. Dale W. Wahlstrom at the Wright Laboratory Air 
Base Technology Branch, Tyndall Air Force Base, Florida, for fabricating the concrete 
fracture specimens. 

The author wishes to acknowledge the following individuals at the Air Force 
Research Laboratory, Munitions Directorate at Eglin Air Force Base, Florida, for their 
contributions listed below: Ms. Voncile Ashley for her assistance with the quasistatic 
splitting tension cylinder tests conducted on the servohydraulic load frame. Mr. Thaddeus 
Wallace for support in scheduling tests and arranging computer equipment for data 
transfer and analysis. Mr. Harold Gilland for providing photographic and electronic 
imaging support. Acknowledgment is made to Dr. Joseph C. Foster for his compelling 
encouragement to pursue higher education. 

Finally, the author acknowledges the support of the Air Force Research 
Laboratory for providing the opportunity to accomplish professional improvement that is 
commensurate with personal goals. 



in 



TABLE OF CONTENTS 

page 

ACKNOWLEDGMENTS iii 

ABSTRACT vii 

CHAPTERS 

1 INTRODUCTION 1 

Objective \ 

Background ] 

Strain Rate Dependency \ 

Fracture Mechanics 2 

Previous Related Dynamic Fracture Studies 3 

Approach 6 

2 NONLINEAR FRACTURE MODELS 8 

Effective Elastic Crack Theory for Nonlinear Fracture Mechanics 8 

The Two-Parameter Fracture Model 8 

The Size Effect Model 12 

Relationship Between Fracture Model Parameters 14 

3 SPECIMEN DESIGN AND SELECTION 16 

The Splitting Tensile Cylinder 16 

Fracture Specimens lg 

Specimen Design for Improved Results 21 

Fabrication of Fracture Specimens 22 



4 



FRACTURE EXPERIMENTS 24 

Quasistatic Fracture Experiments 24 

The Servohydraulic Apparatus 24 

Holed-Notched Cylinder Fracture Tests 25 

Dynamic Fracture Experiments 30 

Split Hopkinson Pressure Bar 30 

Solid Cylinders in Splitting Tension 33 



IV 



Holed-Notched Cylinder Fracture Tests 34 

Ultrahigh-Speed Digital Photography 39 

5 FINITE ELEMENT ANALYSIS 44 

Calculation of the Energy Release Rate 46 

Setup of the Dynamic Analysis 48 

Mesh and Element Selection 50 

Stability and Sensitivity Calculations 51 

Numerical Results of the Fracture Experiments 56 

Summary of the Finite Element Analysis 65 

6 EVALUATION OF FRACTURE PARAMETERS 66 

Fracture Parameters using the Two Parameter Fracture Model 66 

Application of the Two Parameter Fracture Model 66 

Results of the Two Parameter Fracture Model 72 

Fracture Parameter using the Size Effect Method 75 

Application of the Size Effect Method 75 

Results of the Size Effect Method 76 

Relation to Linear Elastic Fracture Theory 80 

Summary on the Application and Results of the Fracture Models 83 

7 SUMMARY AND CONCLUSIONS 84 

Nonlinear Fracture Theories 84 

Fracture Experiments 85 

Finite Element Analysis 86 

Behavior of the Fracture Response 86 

Future Research 87 

Concluding Remarks 88 

APPENDICES 

A COEFFICIENTS FOR CRACK TIP OPENING DISPLACEMENT 

PROFILES 90 

B INCIDENT STRESS CHARACTERIZATION OF THE SPLIT 

HOPKINSON PRESSURE BAR 93 

C DYNAMIC STRENGTH RESULTS OF HOLED-NOTCHED 

CYLINDERS 95 

D SYNCHRONIZED STRESS HISTORY AND ULTRA-HIGH SPEED 

PHOTOGRAPHS 99 



E INITIAL CALCULATIONS FOR THE TWO PARAMETER 

FRACTURE MODEL, AN EXAMPLE USING ONE STRAIN 

RATE, (RATE = 2/S) 121 

F SPREADSHEET ANALYSIS FOR THE TWO PARAMETER 

FRACTURE MODEL 131 

G CALCULATIONS OF THE SIZE EFFECT METHOD 145 

REFERENCES 155 

BIOGRAPHICAL SKETCH 158 



VI 



Abstract of Dissertation Presented to the Graduate School of the University of Florida in 
Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 

DYNAMIC FRACTURE PROPERTIES OF CEMENTITIOUS MATERIALS 

By 

DAVID EDWARD LAMBERT 

December 1998 

Chairman: C. Allen Ross 

Major Department: Aerospace Engineering, Mechanics, and Engineering Science 

An experimental technique with associated analysis method was developed for 
obtaining the material parameters characterizing mode I, dynamic fracture instability for 
cementitious materials. Fracture specimens in the form of notched-cavity splitting tension 
cylinders were failed under dynamic loading of a split Hopkinson pressure bar. Specific 
stress-intensifying cavities of various geometries were created along the cylinders' axis to 
provide structural defects where the cracks would be known to initiate from and to 
provide characterized stress intensity functions, a priori. Quasistatic tests were conducted 
on a servohydraulic load frame and the results provide a comparison basis for the dynamic 
tests. A total of 150 concrete specimens were fractured involving cylinders of 76 mm, 102 
mm, and 152 mm in diameter. 

Fracture parameters were extracted by the application of two nonlinear fracture 
mechanics models, the Two-Parameter Fracture Model and the Size Effect Method. A 
unique application of both models provided an analysis method that required only the 



vu 



measurement of peak loads sustained by the specimen at the transition regime of stable to 
unstable crack propagation. Results show an increase in the apparent fracture toughness 
during dynamic loads corresponding to specimen strain rates in the range of 1 /s to 10 /s. 
This increase in the resistance to fracture is synonymous with the increase in tensile 
strength over the same strain rate regime. The fracture process was observed to remain 
nonlinear throughout all strain rates. The two models provided independent confirmation 
of results. 

Primary instrumentation and diagnostics included electrical resistance strain gages 
for stress interpretation, ultrahigh-speed digital macrophotography for observing crack 
propagation, and finite element analysis for numerical investigations. Finite element 
calculations were conducted to verify the experimental configuration and address inertial 
contributions within the dynamic energy partitioning process at the crack tips. The 
calculations also provided critical insight to the history of the stress intensity and the 
results were synchronized with the specimen stress history of the experiments and the 
photographic record capturing the fracture event. The ability to synchronize the fracture 
event, experiment diagnostics, and finite element calculations proved to be a unique and 
critical tool for the analysis process. 



Vlll 



CHAPTER 1 
INTRODUCTION 

Objective 

The objective was to develop a methodology to obtain the material properties 
characterizing the mode I, opening resistance to fracture for cementitious material 
subjected to dynamic failure loads. Specifically, nonlinear fracture mechanics parameters 
of critical stress intensity, K Jc , and critical crack tip opening displacement, CTOD c , were 
determined for concrete fracture specimens subjected to quasistatic and stress wave 
loading. The technique developed within this dissertation includes experimental design, 
fracture specimen selection and characterization, application of nonlinear fracture 
mechanics, and investigation by finite element calculations. 

Background 

Strain Rate Dependency 

In general, material strength response depends on the rate of deformation or 
straining. The rate dependency in metals is attributed to dislocation mobility and 
generation, while with concrete the process is more complicated, involving such events as 
the linking of microcracks and aggregate bridging. Typical strength increase of concrete 
in tension and compression is illustrated by Figure 1 . It is postulated that the fracture 
properties of the concrete might also be strain rate dependent. 



7.0 





6.0 


fl) 




t_ 








w 

o 


5.0 


, ■ 




(0 








V) 


4.0 










c- 




*-* 






3.0 


<i> 




i . 








ro 




o 


2.0 


E 




ra 






1.0 


Q 





0.0 



O Tensile Strength, Ref. [1] 

+ Compressive Strength, Ref. [1] 



■tPcPo gfo 




O o 



s 



* m* 



i i i iiiiii i i i mini i i i Mini i i i Mini i i i inn i i i imii i i i iilllj I ! I mill I i I mill 

1E-6 1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 

Strain Rate (m/m/s) 

Figure 1. Tensile and Compressive Strength Response of Concrete (source Ross [1]) 



Fracture Mechanics 

Linear elastic fracture mechanics (LEFM) is applicable to brittle materials exhibiting 
a linear relationship between the applied stress and consequent deformation. LEFM 
provides for a single material property, such as the critical stress intensity factor, K Ic , or 
critical energy release rate, G c , to completely describe crack instability and catastrophic 
failure. Cementitious materials are quasibrittle, having a nonlinear and inelastic 
deformation response to an applied stress. The nonlinearity occurs through energy 
dissipation mechanisms of microcracking, void formation, aggregate bridging, etc. that give 
rise to a fracture process zone ahead of the advancing crack. Nonlinear fracture mechanics 
for concrete consists of modifying LEFM to include additional characterizing parameters 
that adequately describe the failure state. Two characterizing parameters are chosen for 
reporting the results in this dissertation: the stress intensity factor, K K and critical crack tip 



opening displacement, CTOD c . Definitions for nomenclature of mode I opening fracture 
(Figure 2) provide clarification for later discussions. 



COD 




CTOD 



Mo. 



\ 



COD = crack opening displacement 
CTOD = crack tip opening displacement 
ao = initial crack length 
a,. = current crack length 



Figure 2. Definition of Crack Opening Displacement 



Critical stress intensity, known as fracture toughness, K Jc , has long been 
established as a material property for elastic materials. The physical significance of 
CTOD c can be construed from the fact that all materials, especially concrete, contain some 
type of initial flaws. When the material is subjected to an external load, these flaws will 
open and propagate. The opening displacement of these cracks can be linked to the value 
of the CTOD. Stable crack extension occurs before the critical fracture load and, hence, 
the critical value, CTOD c , at that moment defines the instant of critical crack propagation. 
Previous Related Dynamic Fracture Studies 

Review of past and current dynamic fracture studies of quasibrittle material reveals 
a deficiency in available experimental methods and post analysis techniques. A rather 
comprehensive review of dynamic fracture testing is given by Davis [2] for metallic 
materials (i.e., linear elastic and/or nonlinear elastic), but these techniques are not 



applicable to the nonlinear inelastic response of concrete. The following research 
activities are provided as the most pertinent to dynamic fracture of concrete. 

Research by John and Shah in 1986 [3] showed an increase in the stress intensity 
function with an increase in crack propagation velocity. Crack velocity was modified by 
increased impact velocity of a Charpy anvil. While explicit fracture parameters were not 
extracted, they do comment on the strain rate sensitivity of CTOD and reinforce the strain 
rate sensitivity of material strength, as seen earlier in Figure 1. 

Additional investigations by Shah [4] in 1990 utilized a Charpy impact device to 
extract rate sensitivity of fracture parameters. However, the analysis assumed that 
fracture toughness was rate independent while CTOD c was modeled as an exponential 
decay with strain rate. Maximum strain rates of only 0.5 /s were achieved. 

Fracture properties were found by Oh [5] to increase with loading rate for 
concrete specimens that were dynamically loaded. Results were presented in terms of the 
displacement rate of the loading actuator and not specimen strain rate. Specimen strain 
rate is the parameter selected to illustrate dynamic effects for the current investigation. 
Oh found that the dynamic fracture energy increased to approximately 1 .4 times that of 
the static value. 

The use of stress wave loading to achieve higher strain rates during fracture was 
validated by Duffy and Shih [6]. General guidance for specimen length dimensions was 
given with respect to characteristics of the loading time and inherent elastic wave speeds. 
The design information validated the analysis assumptions for brittle and ductile fracture 
parameters. Also, ceramic and ceramic matrix composites were shown to have a higher 



dynamic fracture toughness than the fracture toughness obtained during quasistatic 
experiments. 

In related efforts using viscoplastic materials, Rittel and Maigre [7] showed that 
the dynamic fracture toughness of PMMA increased up to eight times the value at lower 
strain rates. Results were reported in fracture toughness versus rate of change in stress 
intensity. Another study using PMMA, by Bacon et al. [8], shows up to a threefold 
increase of dynamic fracture toughness. The Bacon paper also presented criteria for the 
inclusion of inertial forces encountered during dynamic stress wave loading. 
The preceding efforts and associated research highlight the following: 
A deficiency in experimental apparatus to conduct well-controlled fracture events 
at significantly high strain rates. Testing devices such as Drop Tower and 
Charpy Impact have complex stress wave reverberations and inertial 
contributions from the impacting tup. Candidate loading devices should 
maintain known stress wave histories and interaction with the specimen. 
The highest strain rate reported for concrete fracture was 0.5 /s, which, according 
to the strength data of Figure 1 is just at the beginning of significant strain rate 
effects. Therefore, a technique is required to produce fracture parameters in 
specimens under strain rates up to at least 10 /s. 
No consensus exists on the behavior of fracture parameters with high strain rate 
and/or high loading rate. A national standard does not currently exist for 
measuring the fracture properties of concrete, neither for quasistatic nor 
dynamic loading. 



6 

The instrumentation and diagnostics of high rate test apparatuses are typically 
incapable of delivering the input requirements of nonlinear fracture models. 
For example, crack opening measurement gages do not survive the violent 
rupture process of a specimen fractured by typical impact loading. 
Inertial effects during dynamic fracture events can significantly influence the 
fracture process. However, it is possible to design experiments with 
characteristic loading times and specimens length scales that minimize or 
eliminate the inertial effects. 
All fracture is dynamic by nature, because it is a rate process in which a 
discontinuity by voids is created in previously intact material. The addition of the dynamic 
domain introduces the onus of not only re-addressing applicability of stress intensity-based 
fracture theory; but also, it forces examination of suitable experimental apparatus and 
specimen designs for capturing the fracture process and quantifying the time-dependent 
material parameters. The definition of "dynamic" is arbitrary and must be defined with 
respect to time and length scales of the event. A common discriminator is the time of load 
duration compared to the time required for a stress wave to run a characteristic length, say 
crack length. The experimental apparatus and specimen design for this dissertation takes 
into account the criticality of the loading time and dependency on characteristic length. 

Approach 

This investigation seeks to expand on previous dynamic fracture research by 
considering the prior highlighted issues. The approach taken for the experimental 
technique was to incorporate characterized fracture specimens into the stress wave loading 



environment of a split Hopkinson pressure bar. Characterization included initial material 
constituents and engineering properties as well as initial stress intensity functions 
describing the specimen geometry, its boundary conditions, and the externally applied 
forces. In addition, quasistatic data will be generated for the same types of fracture 
specimens to provide baseline properties for the dynamic tests. 

The analysis approach was to identify a nonlinear fracture model that maintains 
data input requirements compatible and/or amenable to data types available from the 
experimental technique. The stress wave loading precluded instrumentation of the 
specimens with fracture measurement devices commonly used in static fracture research. 
Two nonlinear fracture models were investigated and applied to the quasistatic and 
dynamic data. Application of two models provided independent confirmation of results. 
A unique implementation of the models was made so that only the maximum load 
sustained during crack extension was required from the experiment. 

The study was validated through experiment instrumentation and finite element 
analysis. Ultrahigh-speed digital photography was synchronized with stress measurement 
gages to provide insight to the fracture process and stress history leading up to crack 
initiation and subsequent catastrophic specimen failure. Finite element analysis was then 
integrated with time-resolved experimental stress data and photographs to verify the stress 
intensity transients and interpret the state of fracture. The triad of diagnostic tools 
provided corroboration of results and improved understanding of the dynamic fracture 
technique. 



CHAPTER 2 
NONLINEAR FRACTURE MODELS 



Effective Elastic Crack Theory for Nonlinear Fracture Mechanics 

Classical linear elastic fracture mechanics (LEFM) produces a stress singularity at 
the crack tip. Real materials, especially concrete, yield or fail at specific stress levels and 
do not support the singularity. Therefore, LEFM is modified for quasibrittle materials to 
account for energy dissipation mechanisms. Two energy dissipation theories are 
predominant in the fracture mechanics community. An effective elastic crack approach 
assumes that the energy to create new surfaces is much larger than the energy required to 
separate them. By contrast, the fictitious crack approach assumes just the opposite. The 
effective elastic approach assigns an effective crack length to represent the physical crack 
plus an increment to represent the characteristics of the weakened material in the fracture 
process zone. Of the numerous effective elastic models [9-12], the Two-Parameter 
Fracture Model and the Size Effect Method are selected for application to the results 
obtained in this research. A review of both models is made to demonstrate their influence 
on specimen selection and the requirement for varying specimen size and geometry of the 
stress-intensifying cavity. 
The Two-Parameter Fracture Model 

The Two-Parameter Fracture Model (TPFM) of Jenq and Shah [11] characterizes 
failure using the parameters of critical stress intensity factor, jS* fe and the critical crack 



tip opening displacement, CTOD c . Both are required to uniquely define the onset of 
unstable crack growth. The values are defined by Equation (1) in terms of the applied 
stress, material constants, and geometry-specific stress field functions 

K Ic =VNc4™c F ( a c) 



(1) 



C TOD c =^^^H 
c £ , 



^ ; 

\. a c J 



Kj c Critical Mode I Stress Intensity, MPa-m 1/2 

onc Maximum Nominal Stress, Pa 

a Initial Crack Length, m 

a c Critical Effective Crack Length, m 

a Ratio of the Current Effective Crack Length to Specimen Size 

etc Ratio of the Critical Effective Crack Length to Specimen Size 

E Elastic Modulus for Plane Strain, E - ^/(1-v 2 ) 

v Poisson's Ratio 

F(a q ) Geometry Dependent Stress Field Function 

H(ao/a c , a<) Geometry Dependent Function 

Tang et al. [13] incorporated the TPFM within a statistical framework to 
determine the two parameters from only the measured maximum loads, called the peak 
load method. The method was proposed, in part, because of limitations in the method of 
test as recommended by the International Union of Testing and Research Laboratories for 
Materials and Structures (RBLEM). The peak load method eliminates RILEM's 
requirement of closed-loop testing and CTOD instrumentation. Stress wave loading by a 
split Hopkinson bar does not allow for feedback control, but does capture loading history 



10 

and peak load. Also, the inability to measure CTOD on the dynamic and rather violent 
SHPB apparatus make the peak load method a valuable and, in fact, critical tool. A brief 
review of the TPFM via the peak load method is given. 

If the TPFM is applied to two different specimens, then each specimen presents two 
equations in each parameter: 

CTOD 1 (cT l Nc ,al)=CTOD c 

l( 2 l\ S ® 

K lVNo a cr K Ic 

CTOD 2 (cr 2 Nc ,a 2 c )=CTOD c 

Numeric superscripts indicate specimen 1 or 2, superscript S is for Shah's TPFM 
model [11], and the subscript c indicates "critical" value of unstable crack propagation. 
The parameter a c is the critical effective crack length, representing not only the physical 
crack of discontinuous material, but also a length dimension that represents the region 
nonlinear effects, called the fracture process zone. 

If the nominal strengths for each material are known, then the system is 
reduced to four equations with four unknowns; a' c , c? c , A?/ c , and CTOD c . The relations 
are nonlinear, but a procedure is defined to efficiently solve the simultaneous system. 
The relations for the first specimen become 

K ic= K iM) 

(3) 
CTOD c =CTOD c (a l c ) 



11 

The combination of these two equations can be considered as the parametric form 
of a function with a\ as the parameter. So, the function for specimen 1 is of the form 

CTOD c = f l (K? c ). (4) 

The same relationship can be written for specimen 2 if its nominal strength is known. 

CTOD c = f 2 (Kl). (5) 

The two relations are independent of each other because they were obtained from 
two distinct specimen geometries (and/or sizes). Solution of the system is found at the 
intersection of the two CTOD c -f^i c curves defined by Equations (4) and (5). 

Inherent random error and variability of concrete properties make it improbable to 
obtain "exact" measurements of the nominal strength; thus, the solution of the two 
functions may not be an explicit intersection. Tests of at least three distinctly different 
specimen types are used to increase the confidence of the solution. The peak load 
method simplifies the problem to a one-variable optimization problem taking K s Ic as the 
single variable, with CTOD c determined by the form of Equation (2). The single-variable 
minimization is then described by Equation (6). The existence of a unique minimum in 
the least-squares function defines the material property, X s * Ic . The property CTOlf* c is 
then easily determined by evaluating the CTOD avg c relation at A^* /c . 

®(K? c ) = ±[{CTOD c ) av8 -(CTODjJ (6) 

(=i 

n Number of Specimen Sizes or Groups 

CTOD c avg Average CTOD of all Groups at the same K Ic Value 

CTODJ CTOD Values for the / th Specimen 



12 

If all specimens are identical in geometry and size, then the differences in the 
CTOD c -K S j c curves are caused by randomness and measurement errors only. The test 
specimens should be so distinct that the differences between curves are not significantly 
influenced by the errors. The specimens selected for this investigation, given in the next 
chapter, are based on this premise to create several groups of significantly distinct 
specimens. 
The Size Effect Model 

The Size Effect Method (SEM) of Bazant and Kazemi [12] is based upon Bazant's 
size effect law [14] and assumes the fracture energy dissipated at failure is a smooth 
function of structural dimensions and size of the fracture process zone, as shown in Figure 
3 and described by Equation (7). The model defines critical energy release rate, G/, and 
critical effective crack extension, c/, as material constants for an infinitely large specimen. 
The process zone of an infinitely large specimen is able to fully develop without 
interference from a boundary and, hence, c/is a material constant. Additionally, when the 
initial crack length is sufficiently large with respect to c f , then LEFM applies and G/is also 
a material constant. 

Plasticity 
log^^,) (strength criterion) 

\ ^ Linear Elastic 

fofl(Ok)1 iL-----^- Fracture Mechanics 



log(D) 

Figure 3. Size Effect Method 



°N 



13 



EG f 



(7) 



g'(a )c f +g(a o )D 
where on is the nominal strength of the specimen defined in strength criterion as 

c„ Arbitrarily Defined Constant of Specimen Geometry 

P Maximum Load, Pa 

t Specimen Thickness, m 

D Specimen Dimension (e.g., cylinder diameter), m 

a Ratio of Current Crack Length to Specimen Size 

Oo Ratio of Initial Crack Length to Specimen Size 

g((Xo) Combination of Previous Terms and Defined in Equation (9) 

g '(ceo) Derivative of g(a) with respect to a when a = GCo. 

g(a) = naclF 2 {a) (9) 

Material properties can now be found from regression analysis of nominal failure 
stresses from several geometrically similar specimens of different sizes. A linearized 
form of SEM, shown in Equation (10), was recommended by the International Union of 
Testing and Research Laboratories for Materials and Structures (RILEM) [15] 



g'(°o)°N 



x = gM D 

7 = AZ + C with: &{<*•) 

0,-JL 

1 AE 

C 

c f = — 

f A 



(10) 



14 

The material properties, G/and Cf, are now determined by measuring the 
maximum loads of several geometrically different specimens. Direct application of the 
SEM and TPFM models is now possible by simple measurement of the peak loads 
sustained by characterized fracture specimens. The ability to reduce the problem to that 
of measuring peak loads is an integral and necessary part of the ability to transition the 
fracture experiments into the dynamic environment of the split Hopkinson pressure bar. 

It is important to point out that the size effect law was developed by Bazant [14] 
under the constraint that specimens should maintain a proximity to the plasticity regime, 
described by the brittleness number, /?, of no less than 4. The brittleness number, as 
defined in Equation (11), depends on not only the material fracture parameter, c fi but also 
the structural geometry function, g(a). 

g (pc)c f 



Relationship Between Fracture Model Parameters 

Relationships exist between the fracture parameters of the Two-Parameter 
Fracture Model and the Size Effect Method [16]. Parameters K s Ic and G/can be directly 
related to each other for the infinitely large specimen through LEFM. 

<V=£f£ (12) 

Establishing a relationship between the c/and CTOD c is not so direct. The 
specific form of the relation is dependent upon specimen size and characteristic crack 
lengths that define applicability of LEFM. In Bazant and Kazemi's original work [12], 



15 

fracture quantities were derived within the framework of specimens geometrically scaled 
to an infinitely large dimension; but, associated with the infinite size comes the 
implication of infinite initial crack length a . Here, the requirement is relaxed to allow 
for an adequate LEFM criterion when specimens have the initial crack length much larger 
than the effective crack extension quantity, a »Cf. This criterion, Equation (13), 
provides the second approximating relationship between the two fracture models. 



32G f c f 
CTOD c ^-J+ (,3) 

Analysis of the experimental results includes calculation of the crack extension 
values so that the degree of linear elasticity can be quantified. Quantities have now been 
defined for both models and relationships established between them to provide 
independent confirmation of the experimental results. 






CHAPTER 3 
SPECIMEN DESIGN AND SELECTION 

The fracture specimens were selected as cylinders with stress-intensifying cavities 
manufactured along their axes. The intent of preexisting cavities was to provide for a 
material flaw with stress intensity significantly greater than that of the microflaws within 
the concrete. The preexisting cavities create structural defects with known stress 
intensities and sites for crack initiation and propagation. The cavity localizes failure to a 
specific region for diagnostic observation and provides for characterized stress intensity 
functions, a priori, for calculation of fracture quantities. 

Overall dimensions were selected to provide adequate length scales with respect to 
aggregate size of the concrete, to conform with size variations required of the nonlinear 
fracture models, and to maintain consistency with available stress intensity functions. 

The Splitting Tensile Cylinder 

The specimen geometry is based on the splitting tensile cylinder geometry 
commonly used for indirect measurement of concrete tensile strength. The method was 
introduced by a Brazilian named Fernando Carneiro and is documented in ASTM C496, 
"Standard Test Method for Splitting Tensile Strength of Cylindrical Concrete Specimens." 
A cylindrical concrete specimen is positioned such that its longitudinal axis lies 
perpendicular to the stroke actuator of a compression device, leading to diametric 
compression. Specimen and loading configuration is shown in Figure 4 . Narrow load 



16 



17 

bearing strips are placed between the specimen and loading platens to prevent localized 
compression failure at the load contact point. The load is then applied and increased until 
failure occurs by splitting along the axis of the specimen. Stress distribution within the 
specimen follows a Boussinesq (1892) derivation as given by Timoshenko and Goodier 
[17], described by Equation (14) and orientation shown in Figure 5. 





typical failure, 
lines of break-up 



Figure 4. Splitting Tensile Cylinder and Loading Configuration 




Figure 5. Stress Nomenclature for the Splitting Tensile Cylinder 



18 



Oy 
P 

D 
L 

y 



°z = 



ip 

nLD 



o 



IP 



y 



7±D 



D* 



y{D-y) 



-1 



Tensile Stress, Normal to the Loading Direction, Pa 

Compressive Stress, along the Loading Direction, Pa 

Applied Compressive Load, N 

Cylinder Diameter, m 

Cylinder Length, m 

Point of Interest along the Cylinder Diameter, m 

Fracture Specimens 



(14) 



The splitting tensile cylinder specimen is transformed into a fracture specimen by 
modifying the geometry to include a central cavity along the longitudinal axis. The cavity 
design selected for this research is termed a holed-notched cavity (Figure 6). 




Figure 6. Holed-Notched Splitting Tensile Cylinder 



The cavity presents a steep stress field gradient at the notch tips that are 
superimposed on the uniform tensile stress field along the diameter. The state of stress 
presented results in a pure Mode I, opening mode, plane strain fracture. The resultant 



19 

stress field from such a cavity can be found in analytical form for very simple geometries, 
but must be numerically determined for complex geometries, such as the holed-notched 
cylinders. The geometry-dependent function of Equation (1), F(a), is the term that 
captures the stress intensity from such cavities. Geometry functions, F(a), for specific 
combinations of the ratio of the hole radius to specimen radius, r/R, are made available by 
Yang et al. [18]. The functions are found in finite element methods by first calculating the 
energy release rate, G, for a variety of initial notch lengths. Then, using Equation (1) and 
a LEFM relationship between stress intensity and energy release rate, the geometry 
function of a specific initial crack (notch) length is found, Equation (15). 



\naa N 

The geometry function at any notch length is found by regression over the range of 
discrete initial notch lengths. Additional characterizing constraints of the function are a 
fixed hole-to-specimen radius ratio, r/R, and fixed load distributed width, 2t. Stress 
functions for holed-notched specimens used in this investigation are given in Equation (16) 
for r/R=QM and Equation (17) for r/R=0. 12 with both having load distributed width ratio 
of t/R = 0. 16. Both functions are plotted in Figure 7 to show the increase in stress 
intensity by simply changing hole size and notch length. 



_., 0.129- 0.039a + 1.49a 2 -3.394a 3 + 1.802a 4 

F (a) = = (16) 

a(l--a) 2 



_, , 0.255 -0.713a + 3. la 2 -5.16a 3 + 2.52a 4 

F(a) = (n) 

a(l-a) 2 



20 




r/R = 0.08, (Reference 18) 



i i i i | i i i i | i i i i i i i i i | i i i i | i i i i i i i i i i i i i i 



0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.6 

Total Notch Length / Specimen Diameter (2a/2R) 



Figure 7. Stress Intensity Functions for Holed-Notched Specimens 



Numerically derived expressions also exist for the H(aja, a) function in the CTOD 
relationship of Equation (1). These are found in Equation (18) for r/R=0.0S and Equation 
(19) for r/R=0.\2 cylinders. 



H(%,a) = vfy 



0.046 + 2.75a - 3.28a 2 + 4.6a 3 + 0.7a 4 



a 



v a v a ' 



- 0.472 + 6.2a - 8.82a 2 + 10.86a 3 + 2.52a 4 



(1 + a)a 



where the common V(a</a) function is expressed as 



a r 



f„ \ 2 



a a 



a, 



K a ) 



+ L-. 



a, 



a 



(18) 



(19) 



(20) 



with coefficients, Z,,, given in Appendix A as a function of cylinder and notch dimensions. 
Now, all equations exist for determining the state of stress for a selection of holed- 
notched cylinders and for evaluating the fracture parameters. The following section 



21 

presents a method to select specimen dimensions, r/R and a,/R, that minimizes physical 
measurement error and improves the regression analysis for both of the models. 

Specimen Design for Improved Results 

The peak load method of the TPFM and the SEM take advantage of specimen 
distinction; that is, specimens of different size, geometry, and/or cavity design. The stress 
intensity function, F(a), quantifies those design variables. Solution confidence is 
increased in the TPFM as the intersections of the K Ic -CTOD c curves approach a single 
point. Solution confidence of the SEM is increased with regression of specimens with 
largely different abscissa values, X, meaning specimens of large differences in size or stress 
intensity function, F(a). Physical size limitations are placed on specimens appropriate for 
dynamic testing, in this case, the split Hopkinson pressure bar. Thus, geometry is the 
remaining free parameter for achieving vast specimen distinction. Geometry is represented 
by the dimension ratios r/R and aJR of the stress-intensifying cavity. 

Four distinct holed-notched cylinder designs were selected for fracture 
experiments of this investigation. Additionally, the four types were manufactured at three 
size scales: 76-mm (3-inch), 102-mm (4-inch), and 152-mm (6-inch) outer cylinder 
diameter. Table 1 lists the specimens selected. Selection of the specific cavity ratios 
follows the recommendations made by Tang et al. [19] for maximizing the regression 
accuracy of the nonlinear fracture models. 

The size variations and holed-notch cavities of Table 1 provide a brittleness 
number, described in Equation (11), of approximately /? = 5 to provide a sufficient 
applicable range of the nonlinear model. 



22 
Table 1 . Specimen Characteristics 



Diameter 


Type 


Hole radius/Specimen Radius 


Initial Notch/Specimen Radius 


D 




(r/R) 


(a</R) 


76, 102, 152 mm 


Type 1 


0.08 


0.25 


76, 102 mm 


Type 2 


0.08 


0.45 


76, 102 mm 


Type 3 


0.12 


0.25 


76, 102, 152 mm 


Type 4 


0.12 


0.45 



Fabrication of Fracture Specimens 

All cylinders were cast from the same batch of concrete. The limestone aggregate 
was sieved to pass a 9.5 mm (3/8 in) sieve. Mix proportions for the concrete are listed in 
Table 2. The holed-notched cavities were created in two steps. The holes were created 
during the casting process by inserting a stainless steel rod of desired hole dimension, r, 
along the casting die axis. Location of the rod was maintained by holes in the end of the 
mold and its cover plate. A petroleum-based release agent was used on the rods to ease 
their extraction after one day of curing. The cylinders were also removed from the mold 
at this point and then cured underwater for 28 days. The cast cylinders were long enough 
to produce multiple specimens by cutting with a diamond-tipped saw. All holed-notched 
cylinders were wet cut to 38 mm (1 .5 in) length during the 28 day underwater cycle and 
were promptly returned to the water bath. After two weeks of drying, the notches were 
cut with a carbide-tipped reciprocating saw. Final notch lengths were measured using a 
Deltronics® model DH214 optical comparator with a 20 times magnification lens. All 
specimens cured in air for at least 90 days before testing. 



23 



Table 2. Concrete Mix Proportions 




Component Quantity (kg) 


Quantity (lbs) 


9.5 mm Max. Limestone Aggregate 1 12.86 


248.6 


Portland cement 24.56 


54.1 


Sand 88.30 


194.5 


"F" Fly Ash 16.16 


35.6 


Water 24.56 


54.1 



Characterization of physical and engineering properties of the concrete was 
accomplished through tests of solid cylinders, 102 mm (4 in) diameter by 203 mm (8 in) 
length and 152 mm (6 in) diameter by 304 mm (12 in) length, cast from the same batch of 
concrete used for the holed-notched specimens. The mass density, w, of the concrete was 
measured as 2280 kg/m 3 (142.2 lb/ft 3 ). The modulus of elasticity, E, was determined from 
the empirical equation developed by the American Concrete Institute (ACI), Equation 
(20), using the unconfrned static compressive strength. 

E = 33w*Jf! (20) 

E Quasistatic Elastic Modulus (MPa) 

w Mass Density (kg/m 3 ) 

f' c Unconfrned Compressive Strength (MPa) 

Two cylinders of each size were tested in compression in a Forney® System 2000 
(Forney, Incorporated, Wampum, Pennsylvania) load frame. The failure stress of the four 
cylinders provide an average unconfrned compressive stress of/' c = 37.85 MPa (5491 
psi), resulting in a modulus of£ = 28.6GPa (4.15xl0 6 psi). 



CHAPTER 4 
FRACTURE EXPERIMENTS 



Quasistatic Fracture Experiments 

The Servohydraulic Apparatus 

An Instron® Series 1332 servohydraulic machine at the Air Force Research 
Laboratory, Munitions Directorate, was used for quasistatic testing. The system is a 
closed-loop device for applying either load control or displacement control up to 
maximum limits of 244 kPa (50,000 lbs) and 50 mm (2 in), respectively. A basic 
description of the apparatus includes the load frame for housing the load cell, actuator, 
loading platens, and hydraulic pump. The load cell contains a resistance strain gage array 
to provide electronic signals proportional to the force applied to the sample. An actuator 
is mounted to the underside of a fixed table of the load frame and has a servo valve to 
control the flow of oil from the pump to the actuator and, hence, the force applied to the 
specimen. Displacement control is achieved through a Linear Variable Differential 
Transformer (LVDT) mounted in the actuator. The LVDT provides a voltage signal 
proportional to the piston displacement. An Instron series 8500 controller monitors the 
prescribed load and displacement constraints for closed-loop feedback processes. 

Data acquisition was made on a Nicolet® Pro 20 transient digitizer with four 
channels at 16-bit resolution. The load and displacement signals were recorded at a 
sample interval of 0.2 s with a total temporal duration of 100 s (minimum). The strain 

24 



25 



gage and LVDT voltage signals were cast into load and displacement values, respectively, 
through proportionality constants assigned in the controller prior to test. 

Stability of crack growth depends on the rate of change in the energy release rate, 
G. In turn, the energy release rate is proportional to the applied load. Although the 
driving force is the same for both load control and displacement control, the rate of 
change of the driving force depends on how the structure is loaded. Displacement control 
typically maintains a more stable crack growth process than load control and, hence, was 
used in the quasistatic fracture experiments. 

Displacement controlled tests, by definition, require a priori knowledge of the 
displacement rates that the controller prescribes to the actuator throughout the 
experiment. There is no explicit relationship between the actuator displacement (or 
velocity) and the resulting splitting stress (or stress rate) for the nonlinear material. 
Therefore, in order to obtain an approximate displacement rate, a preliminary test series 
with one cylinder of each diameter was conducted under load control of 17.24 kPa/s (150 
psi/min). The 17.24 kPa/s rate was the median of a stress rate range, 1 1.5 to 23 kPa/s 
(100 to 200 psi/min), recommended by ASTM 496 for conducting splitting tension tests 
using solid cylinders. The resulting nonlinear displacement records were fit to constant 
rates (given in Table 3) to be used as the control parameter in the displacement controlled 
fracture tests. 
Holed-Notched Cylinder Fracture Tests 

A total of thirty-three (33) holed-notched specimens and three (3) solid cylinders 
were fractured under quasistatic loading. These thirty-six tests to provided baseline 



26 



Table 3. Actuator Displacement Rates from Load Control Tests 



Specimen Diameter 


Actuator Displacement Rate 


mm (in) 


mm/s 


76.2 (3.0) 


0.001405 


101.6 (4.0) 


0.001702 


152.4 (6.0) 


0.002304 



fracture characteristics of the concrete and comparison quantities for the dynamic test 
results. The solid cylinders, 76 mm (3 in) diameter by 38 mm (1.5 in) long, provided basic 
material characterization of the splitting tensile strength. All three cylinder sizes and all 
four cavity types of the holed-notched cylinders found in Table 1 were tested. At least 
three tests of each specimen type were performed to mitigate any anomalies produced by a 
single specimen. The tests utilized the displacement rates of Table 3 to maintain a stress 
rate between 1 1 .5 to 23 kPa/s (100 to 200 psi/min). 

The specimens were mounted between the 76 mm (3 inch) diameter loading 
platens with the notch aligned along the direction of loading. The load distributors were 
machined from ASTM 1018 mild steel to adapt the curved diameter of the specimen with 
the flat surface of the loading platens, as per earlier discussions in Chapter 3. The load 
distributors had scaled dimensions of a constant 2t/D = 0.16 (Figure 4) for each diameter. 

Displacement and load plots from every test are not given in order to maintain 
brevity. A typical example showing the linearity maintained in the displacement record is 
given in Figure 8 for a 102 mm diameter cylinder with a cavity described by a/R = 0.45 
and r/R = 0.08. The corresponding stress plot, being proportional to the load plot via 
Equation (14), is given in Figure 9. Average stress rate of the applied load, up to the 



27 

point of crack initiation, was calculated by regression of the linear portion to ensure it was 
within the desired range of 1 1 .5 to 23 kPa/s (100 to 200 psi/min). The three solid cylinder 
tests had an average splitting tensile strength of /, ' = 3.30 MPa (479 psi). 

0.50 



E 


0.45 


E 


0.40 


■*-> 




c 

CD 


0.35 


E 

0) 


0.30 


o 




CO 


0.25 


Q. 




CO 


20 


u 




o 


0.15 


•4—> 




CO 

-5 


0.10 


*•{ 




a 


0.05 



0.00 




20 40 60 80 100 120 140 160 180 200 220 240 260 

Time (sec) 



Figure 8. Actuator Displacement Record from a Quasistatic Test 



3.00 



CO 




Q. 


2.50 


^-" 




CO 




to 

CI) 


2.00 


.b 




CO 




o 


1.50 


CO 




c 




(1) 




+J 


1.00 


O) 




c 




*2 






50 


u. 




CO 





0.00 



Peak Stress — v 


Specimen failure ^^_^ -- ^ 


Crack initiation \^ ^^^"^ L^^ 


y 


slope = 17.24kPa/s 




s^ 


(2.5 psi/s) 




^r 




\ \ i I i i i I I I I I i I I | I i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | 


-++- 



20 40 60 80 100 120 140 160 180 200 220 240 260 

Time (sec) 



Figure 9. Splitting Tensile Stress Signal from a Quasistatic Test 



28 



Peak stress and associated strain rate obtained from each of the static fracture tests 
are given in Table 4. Average strain rate was calculated from the slope of the stress rate 
using Equation (21). All strain rates were of the order 10" 8 m/m/s and that number will be 
associated with the static strength values in later analysis of the fracture data. 



e 
& 
E 



s = — 
E 



Average Strain Rate during Specimen Load to Failure, m/m/s 
Stress Rate from Regression of the Stress Curve, Pa 
Elastic (quasistatic) Modulus of the Concrete, Pa 



(21) 



Table 4. Results from Quasistatic Fracture Experiments 



Group # Test Name 



Specimen Hole Notch Strain Rate 

Diameter Diameter Length 

mm (in) mm mm (m/m/s) 



Peak Stress Average Stress 
of Each Group 
MPa(psi) MPa (psi) 



Solid 
Cylinders 


Solid 1 
Solid 2 


75.87 
75.95 


(2.987) 
(2.990) 


N/A 
N/A 


N/A 
N/A 


1.91E-08 
2.19E-08 


3.744 (543) 
3.516 (510) 


3.601 


(522) 




Solid 3 


75.95 


(2.990) 


N/A 


N/A 


2.19E-08 


3.544 (514) 






Group 1 


Group 1-1 


75.87 


(2.987) 


6.35 


18.7 


3.02E-08 


3.896 (565) 


3.482 


(505) 




Group 1-2 


75.95 


(2.990) 


6.35 


19.1 


1.86E-08 


3.448 (500) 








Group 1-3 


75.87 


(2.987) 


6.35 


18.7 


2.35E-08 


3.103 (450) 






Group 2 


Group 2-1 


75.87 


(2.987) 


6.35 


33.0 


1.26E-08 


2.517 (365) 


2.542 


(369) 




Group 2-2 


75.95 


(2.990) 


6.35 


32.7 


1.35E-08 


2.768 (402) 








Group 2-3 


75.95 


(2.990) 


6.35 


33.9 


2.10E-08 


2.341 (339) 






Group 3 


Group 3-1 


75.95 


(2.990) 


9.53 


18.5 


1.58E-08 


3.068 (445) 


3.099 


(449) 




Group 3-2 


75.87 


(2.987) 


9.53 


18.3 


2.56E-08 


2.700 (392) 








Group 3-3 


75.87 


(2.987) 


9.53 


18.8 


2.26E-08 


3.528 (512) 







29 



Table 4--continued 



Group # Test Name 



Specimen 


Hole Notch Strain Rate 


Diameter 


Diameter Length 


mm (in) 


mm mm (m/m/s) 



Peak Stress 
MPa (psi) 



Average Stress 

of Each Group 

MPa (psi) 



Group 4 


Group 4-1 


75.95 


(2.990) 


9.53 


33.0 


1.93E-08 


2.275 


(330) 


2.211 


(321) 




Group 4-2 


75.95 


(2.990) 


9.53 


33.8 


1.73E-08 


2.389 


(347) 








Group 4-3 


75.95 


(2.990) 


9.53 


33.5 


2.10E-08 


2.321 


(337) 








Group 4-4 


75.87 


(2.987) 


9.53 


34.1 


2.24E-08 


1.934 


(281) 








Group 4-5 


75.87 


(2.987) 


9.53 


34.1 


3.08E-08 


2.137 


(310) 








Group 4-6 


75.95 


(2.990) 


9.53 


34.0 


3.48E-08 


2.206 


(320) 






Group 5 


Group 5-1 


101.2 


(3.986) 


8.08 


24.5 


3.14E-08 


3.275 


(475) 


3.264 


(473) 




Group 5-2 


101.2 


(3.986) 


8.08 


25.5 


2.37E-08 


3.472 


(504) 








Group 5-3 


101.2 


(3.986) 


8.08 


25.1 


3.44E-08 


3.046 


(442) 






Group 6 


Group 6-1 


101.2 


(3.986) 


8.08 


45.5 


2.41 E-08 


2.227 


(323) 


2.272 


(330) 




Group 6-2 


101.2 


(3.986) 


8.08 


45.3 


2.62E-08 


2.049 


(297) 








Group 6-3 


101.4 


(3.993) 


8.08 


45.5 


3.81 E-08 


2.539 


(368) 






Group 7 


Group 7-1 


101.4 


(3.993) 


12.7 


24.3 


1.97E-08 


2.849 


(413) 


2.792 


(404) 




Group 7-2 


101.4 


(3.993) 


12.7 


25.7 


3.39E-08 


2.650 


(384) 








Group 7-3 


101.2 


(3.986) 


12.7 


25.6 


3.04E-08 


2.878 


(417) 






Group 8 


Group 8-1 


101.2 


(3.986) 


12.7 


45.8 


3.14E-08 


2.096 


(304) 


2.180 


(316) 




Group 8-2 


101.2 


(3.986) 


12.7 


45.5 


2.22E-08 


2.515 


(365) 








Group 8-3 


101.4 


(3.993) 


12.7 


45.4 


2.64E-08 


1.928 


(280) 






Group 9 


Group 9-1 


151.9 


(5.980) 


12.7 


37.9 


9.15E-09 


3.275 


(475) 


2.969 


(431) 




Group 9-2 


152.1 


(5.988) 


12.7 


38.2 


9.73E-09 


2.849 


(413) 








Group 9-3 


151.9 


(5.980) 


12.7 


37.8 


1.10E-08 


2.784 


(404) 






Group 10 


Group 10-1 


151.9 


(5.980) 


19.1 


68.2 


1.38E-08 


1.965 


(285) 


1.950 


(283) 




Group 10-2 


152.1 


(5.988) 


19.1 


68.4 


1.21 E-08 


2.162 


(314) 








Group 10-3 


152.1 


(5.988) 


19.1 


68.5 


1.24E-08 


1.808 


(262) 







30 
Dynamic Fracture Experiments 

The servohydraulic devices, such as the Instron, are limited in loading rate by 
system response and inertia of their own components. To achieve strain rates on the order 
of 1 /s to 10 /s, an impact or stress wave loading apparatus is typically required. 
Split Hopkinson Pressure Bar 

A widely used stress wave loading device for determining dynamic material 
behavior is the split Hopkinson pressure bar (SHPB) or Kolsky bar [20]. Details of the 
equations and principles of the bar are widely documented for a variety of test types [21- 
23], but a brief summary of the equations as applied to the splitting tensile test is given. 

A 76 mm (3 in) diameter SHPB located at the University of Florida Graduate 
Engineering and Research Center, Shalimar, Florida, was used for this program. A 
schematic of the bar is given in Figure 10. A 762 mm (30 in) long striker bar is 
accelerated by compressed gas into a 3048 mm (120 in) long incident bar to create an 
elastic stress wave approximately 300 |is in duration. This elastic wave propagates down 
the length of the incident bar and reaches the interface of the bar and specimen. Shock 
impedance differences of the metal bar and concrete cause a part of the signal to be 
reflected and only a portion to be transmitted into the specimen. The specimen/transmitter 
bar interface presents a second wave reflection and the transmitted wave represents the 
proportional strain experienced by the specimen. The incident and reflected signals are 
recorded by a strain gage located at the mid-length of the incident bar and the transmitted 
signal is recorded by a gage located at the mid-length of the transmitter bar. Typical stress 
signals from one of the splitting tension cylinder tests are shown in Figure 11. 



31 



Striker Bar 



Incident Bar 



Transmitter Bar 




Initial 
Velocity 



strain gages 



Cylinder in 
diametric compression 



strain gages 



Figure 10. Split Hopkinson Pressure Bar 



CO 

Q. 



120.0 
100.0 
80.0 t 



- 60 
a> 40.0 

CD 

O 20.0 
•| 0.0 

55 -20.0 I 

| -40.0 -_ 

t "60.0 

B 

co -80.0 

W-100.0 



■120.0 



• Incident Stress, MPa 
Transmitted Stress x 10 MPa 



Reflected wave 



Incident wave 





\ Transmitted Wave 



■ i | ' ' ' i | i i i i | i i i i | i i i i | i i i i | i i t i i i i i 



i i i r~ 



-60 50 150 250 350 450 550 650 750 850 950 
Time from Incident Gage Trigger, (s) 

Figure 11. Stress Signals from SHPB Strain Gages 



All strain gage signals were produced from perturbation of a full Wheatstone 
Bridge circuit that was amplified and conditioned using a Measurements Group®, Model 
231 1 conditioning amplifier (Measurements Group, Raleigh, North Carolina). ANicolet* 



32 

4094 digital oscilloscope recorded all gage signals at 8-bit precision and at a 50 ns sample 
rate. The Nicolet was triggered at a nominal -50 mV threshold and utilized a 250 p.s pre- 
trigger memory to guarantee capture the entire incident signal. 

The nominal stress supported by the specimen is found from the maximum 
transmitted stress in the bar and a force balance at the bar/specimen interface. The 
splitting tension Equation (14) is transformed to Equation (22) from the force balance on 
the loaded specimen. Average strain rate during the fracture process is taken as the slope 
of the rising portion of the transmitted signal, as in Equation (23). 

2P Ufa 

" N -^5 ~LD- (22) 

s= i£ < 23) 

on Splitting Tensile Stress, Pa 

Rb Radius of the Split Hopkinson Bar, m 

L Specimen Length, m 

D Specimen Diameter, m 

e Average Strain Rate During the Linear Rise to Peak, m/m/s 

Act Change in Transmitted Stress over Time Interval At, Pa 

E Elastic Modulus from Static Characterization, Pa 

The static modulus was used for all analysis requirements of the dynamic 
experiments. The assumption of rate independence for the elastic modulus of concrete 
was supported by previous researchers [24-27]. A dynamic modulus can not be obtained 
from the slope of the transmitted signal in the dynamic tests because, even though the 



33 

specimen is responding under a constant strain rate the specimen has not yet attained a 
uniform stress distribution along its diameter [23]. The specimen is subjected to a 
multitude of stress wave reflections that incrementally bring the local stress up to a 
constant value across the diameter. 

To achieve different specimen strain rates the impact velocity of the striker bar is 
changed. Striker impact velocity dictates the longitudinal stress imparted to the incident 
bar through the conservation of momentum and energy to give the particle wave jump 
conditions of Equation (24). The rise time and magnitude of the incident stress then 
influences the strain rate of the specimen. Striker bar velocity is proportional to the 
chamber pressure of the gas reservoir launching the striker. The chamber gas pressures 
and the incident stress signals produced are characterized in Appendix B in terms of 
incident signal rise time and magnitude of the stress plateau. 

°I=\pcqV s (24) 

<Ji Incident Stress created by Striker Bar Impact, Pa 

p Mass Density of the SHPB, kg/m 3 

Co Longitudinal Elastic Wave Speed of the SHPB Steel, m/s 

V s Striker Bar Impact Velocity, m/s 

Solid Cylinders in Splitting Tension 

Experiments were carried out on ten (10) solid cylinders (no stress-intensifying 
cavity) to characterize the concrete's fundamental strength response as a function of strain 
rate and to obtain baseline data for comparison to the fracture specimens. These cylinders 
were nominally 76 mm (3 in) in diameter and 38 mm (1 .5 in) long. The cylinders were 



34 

placed between the incident and transmitter bar end surfaces using the steel load 
distributors — as in the quasistatic tests — with dimensions having a width-to-specimen 
diameter ratio of 2t/D = 0. 16. Results from the ten solid cylinders under dynamic loading 
are given in the next section with the holed-notched data. 
Holed-Notched Cylinder Fracture Tests 

A total of 104 holed-notched cylinders were fractured in the SHPB with the 
notches oriented in the direction of applied load, as in Figure 12. The test matrix was 
partitioned to achieve two or three tests of each distinct group at several strain rates in the 
range of 1 /s to 10 /s. The variability in the launch velocity of the incident bar does not 
allow one to precisely achieve a pre-determined strain rate in the SHPB. Generally, strain 
rates increased with the striker bar's impact velocity, but they did so with variability of 
order 2 Is. 



Incident 
Bar 




Transmitt 
Bar 




Incident 
Bar 



Transmitte 
Bar 



Front View 



Top View 



Figure 12. Fracture Specimens in the Split Hopkinson Pressure Bar 



Data analysis of the transmitted signal involved transferring the binary file from the 
recorder to an ASCII format on a personal computer. Strain rate was determined by 
Equation (23) with the slope determined by linear regression of the rising slope of the 
transmitted signal. Grapher® (Golden Software, Inc., Golden, Colorado) software was 



35 

used for all signal processing and curve-fitting routines. The signals contained high 
frequency waves superposed on a baseline signal structure that presented uncertainty to 
what one would call the "peak" amplitude. The noise is well documented in elastic wave 
theory governing cylindrical bar impacts and is a result of wave transmission perturbations 
from imperfect bar/bar and bar/specimen interfaces, as well as, from wave dispersion 
phenomenon inherent of wave propagation through elastic media [27-30]. To circumvent 
data variations induced by the noise and standardize the selection process for the peak 
amplitude, a quadratic fit was made to the peak region of each signal. An example of the 
linear fit to the strain rate and the quadratic fit to the peak is given in Figure 13. Results 
of individual specimens for all of the dynamic experiments are given in Appendix C and 
include time-to-failure of the specimens. Time-to-failure was taken as the time from initial 
transmitted signal to its peak. The peak stress dynamic data was cast into a form more 
suitable for use in the fracture models by curve fitting each of the ten group's response as 
a function of strain rate. 

The fracture models required input in the form of peak stress of each group at the 
same strain rate increment. The data of Appendix C shows a sundry of strain rates with 
only a few tests having identical rates that could be used for comparison. Therefore, the 
data of each group was regression fit to a linear form. Argument over the functional form 
of the fit can be made, but the linear form is defended by clarifying that the interval is only 
a sub-interval, 2 /s to 8/s, of the entire strain rate spectrum. If one obtained additional 
data just below or just above the interval selected, then a piece-wise multi-linear 
description could be added to the data set. 



36 





0.00 


— 




-0.05 


_Z 


S 




- 


<D 


-0.10 


— 


o> 




- 


ro 






O 


-0.15 


— 


g 




z 


ra 


-0.20 


— 


*-» 






CO 




~ 


c 
o 


-0.25 


— 


0) 




_ 


ra 


-0.30 


— 








o 




- 


> 


•0.35 


... 


TJ 






CD 




— 


*2 


-0.40 





b 




_ 


(A 




- 


ra 


-0.45 


— 


\— 






\- 




- 




-0.50 


— 



-0.55 




Time(s) 



Figure 13. Signal Analysis for Peak Stress and Strain Rate 



Results of the ten solid cylinders are given in Figure 14 along with the linear fit. 
Splitting tensile strength of the cylinders is observed to significantly increase to well 
beyond the 3.6 MPa obtained in the quasistatic tests. The graphs of Figure 14, Figure 15, 
and Figure 16 show the individual test data and their accompanying linear fit for the 76 
mm, 102 mm, and 152 mm diameter fracture specimen cylinders, respectively. The 
coefficients of the regression equations, Equation (25), are given in Table 5. 



*%™P # =Ae + B 



(25) 



a Nc Peak Splitting Tensile Strength at Various Strain Rates, MPa 

A Regression Coefficient, Slope of the Strain Rate Relation, MPa/s 

B Regression Coefficient, Intercept of the Strain Rate Relation, MPa 



37 



to 

Q. 



i 

CO 

J*: 
to 
<d 



I^.UU 


+ 


Solid Cylinders, Experiment 
■ Solid Cylinders, Regression Fit 






- 




+ 




10.00- 








- 








+ 


8.00 - 




4r"+ 






fi on 












i 


T~^n ~ i ~^r~ i ~ 


r" T ~T" 


1 1 ' ! 



1.00 2.00 



3.00 4.00 5.00 6.00 

Strain Rate, (m/m/s) 



7.00 8.00 



Figure 14. Dynamic Results of 76 mm Diameter, Solid Cylinders in Splitting Tension 



12.00 



10.00 



CD 

? 8.00 H 



1 

CO 



6.00 
4.00 
2.00 




1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 

Strain Rate, (m/m/s) 



+ Group 1 , Experiment 

O Group 2, Experiment 

H Group 3, Experiment 

A Group 4, Experiment 



Group 1 , Regression Fit 
Group 2, Regression Fit 
Group 3, Regression Fit 
Group 4, Regression Fit 



Figure 15. Dynamic Results of 76 mm Diameter, Holed-Notched Cylinders 






38 



2 






10.00 



8.00 



6.00 



4.00 



2.00 




1.00 2.00 



3.00 4.00 5.00 6.00 

Strain Rate, (m/m/s) 



7.00 8.00 



+ Group 5, Experiment 

O Group 6, Experiment 

■ Group 7, Experiment 

A Group 8, Experiment 



Group 5, Regression Fit 
Group 6, Regression Fit 
Group 7, Regression Fit 
Group 8, Regression Fit 



Figure 16. Dynamic Results of 102 mm Diameter, Holed-Notched Cylinders 



co 



12.00 -n 



10.00 



V) 

to 

_* 8.00 

CO 
CD 
Q_ 



6.00 






o 



o +. -o 



£icr Qd 



1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 

Strain Rate, (m/m/s) 



+ Group 9, Experiment 
O Group 10, Experiment 



Group 9, Regression Fit 
Group 1 0, Regression Fit 



Figure 17. Dynamic Results of 152 mm Diameter, Holed-Notched Cylinders 



39 





Table 5. 


Regression Fit for Stress and Strain Rate Data 




Group 


Diameter 


a/R 


r/R 


Number of 


Slope, A 


Intercept, B 




(mm) 






Tests 


(MPa/s) 


(MPa) 


Group 1 


76 


0.25 


0.08 


13 


0.6456 


5.1119 


Group 2 


76 


0.45 


0.08 


14 


0.4758 


3.3083 


Group 3 


76 


0.25 


0.12 


14 


0.8032 


3.6677 


Group 4 


76 


0.45 


0.12 


12 


0.7321 


1.9323 


Group 5 


102 


0.25 


0.08 


6 


0.6662 


4.6033 


Group 6 


102 


0.45 


0.08 


7 


0.8192 


1.8255 


Group 7 


102 


0.25 


0.12 


9 


0.6993 


3.9877 


Group 8 


102 


0.45 


0.12 


9 


0.7348 


2.2196 


Group 9 


152 


0.25 


0.08 


10 


0.9370 


4.6537 


Group 10 


152 


0.45 


0.12 


10 


0.6586 


4.7640 






Ultrahigh-Speed Digital Photography 

An additional diagnostic included in the SHPB test was an ultrahigh-speed digital 
camera. The Imacon 468® (Hadland Photonics, Cupertino, California) is an eight frame, 
image intensified digital camera capable of imaging at 50 million frames per second. It 
was positioned to capture the fracture process during the fracture event by viewing the 
specimen surface containing the hole-notch (see "Front View" of Figure 12). The camera 
was synchronized with the strain gage signals by triggering the camera with the amplified 
output signal of the conditioning unit. Therefore, the stress and photograph records 
maintained identical temporal reference basis. All tests used either 5 us or 10 \xs 



40 

interframe times with effective exposure duration of 120 ns. A simple wooden box with 
glass optical port was used to protect the camera from dust and debris during the test. 
Illumination was made using a 600 J flash strobe that was triggered 100 u.s before the 
camera began imaging. The exact time of crack initiation was not known before testing of 
a new cavity type or change in cylinder size. Estimates on when to begin photographing 
were made by taking into account the travel time of the elastic wave from the strain gage 
to bar/specimen interface and fracture development times learned as testing progressed. 
The temporal window for photography was extremely narrow. The event depended on 
specimen size, cavity design, and loading rate - - all of which varied throughout the 
research. Typical time for the crack to propagate across the entire samples was only 10 u,s 
and variability of that time between identical samples and loading profile was 
approximately 30 u,s. However, the eight images, obtained at 5 us per frame, provided 
only a 40 u.s window and, thus, the ability to capture the event was not guaranteed. 
Nevertheless, nearly one-third of all tests captured the initial crack propagation and 
progression of the crack. 

The camera provided data critical to post-analysis of the fracture event and also 
verified several questions regarding the experimental technique. The photographs verified 
that the specimen had symmetric crack propagation from both notch tips. The holed- 
notched cylinders contain two notch tips and non-symmetric crack growth would have 
completely changed notch tip stress fields, re-partitioned the energy, and invalidated the 
stress intensity geometry functions, F(a). [13]. The image in Figure 18 shows an example 
of symmetric crack propagation. 



41 




Figure 18. Symmetric Crack Growth During Dynamic Loading (scale increments ■ 1/16 in) 



The photographs also validated the boundary conditions created by the steel load 
distributors. ASTM 496 specifies the use of plywood as the distributor material. 
However, the shock impedance differences between plywood and SHPB steel attenuate 
the transmitted signal to unacceptable levels. The use of steel load distributors, matched 
to the SHPB material impedance, minimizes signal degradation by reducing imperfections 
in the wave transmission. There was concern that the steel distributors would initiate 
fracture at the cylinder's edge instead of at the notch tips. A total of 35 experiments 
produced photographs that captured the specimen/load distributor interface and none of 
them showed fracture occurring in that region. 



42 

Additional information from the photographs was the verification of single crack 
paths. The geometry functions, F(a), were solutions to single crack perturbations of the 
stress field. Multiple cracks would have presented a highly complex stress field to be 
characterized and forced recalculation of the geometry functions. Images show that single 
cracks initiated from the notch tips, but some did have subsequent crack branching. Crack 
branching is not the same as having multiple initial cracks; for branching is included (in 
fact, it is expected to occur) in the nonlinear aspects of the fracture process zone. 

Synchronization of the camera with the gage signals provided detailed insight to 
the fracture process. The transmitted stress signal was matched directly to crack evolution 
and specimen failure through the common reference of time on the incident strain gage. 
The reference point was taken as the time required for the incident stress wave to reach 
the interface of the load strip (on the incident bar side) and the specimen. This reference 
system was also carried in to the numerical analysis section to create a triad of information 
used to investigate and validate the experimental method. An example of one 
synchronized signal and photo data set is shown in Figure 19. Additional stress and 
photograph records are given in Appendix D. Typical of all the stress/photo data was that 
the crack initiated at approximately one-half to two-thirds of the peak amplitude and then 
continued to propagate to its instability point, denoted by the specimen reaching peak 
stress. 

In summary, a significant amount of quasistatic and dynamic strength results have 
been obtained for ten distinct groups of fracture specimens. The test results and technique 
were obtained in a method that provides suitable application to the fracture models. These 



43 

strength results will be used in the calculation of fracture parameters after discussions and 
investigation of the experiment by finite element analysis. 



10.0 




Figure 19. Synchronized Photographs and Transmitted Stress Signal 






CHAPTER 5 
FINITE ELEMENT ANALYSIS 

The experimental effort provided a measure of the overall structural response to 
the dynamic loading. Finite element analysis was performed to achieve greater insight to 
specific phenomena occurring within the specimen that characterized the failure process. 
The Automatic Dynamic Incremental Nonlinear Analysis (ADINA) [31] finite element 
code was used. Specific objectives of the finite element analysis included evaluating the 
contribution of the inertial forces to the energy available to cause fracture, investigating 
time-dependence of the stress intensity caused by the stress wave loading, and examining 
symmetry of the stress intensity at each notch tip. 

The energy release rate, G, and the corresponding mode I stress field were the 
quantities computed for selected holed-notched cylinder designs. The energy release rate 
characterizes the stress intensity function, F(a), and, through the LEFM relationship of 
Equation (26), the functions obtained by Yang et al. [18] were validated for the dynamic 
experiments. This chapter describes how the simulations verified F(a) for a range of 
specimen sizes. The validation process included confirmation that the notch tips 
developed symmetric mode I stress fields and that the magnitude of the energy release rate 
from static loading was equivalent to that attained during dynamic loading near the instant 
of unstable crack propagation. 

r - ( K rf - ™N F ( a f a 

u ~ f " f ( 26 ) 



44 



45 

The basic concept of finite elements is to replace the solid body with an 
interconnected system of sub elements. At these connections it is required that the 
conditions of equilibrium of tractions and compatibility of deformations be satisfied. 
Explicit relationships between nodal tractions and deformations, as well as, internal stress 
distributions and strains are found through known constitutive properties of each material. 
Stress intensity of a crack is then determined from an account of the energy proportioned 
between internal strain energy and kinetic energy with that supplied by the tractions. The 
general solution process can be summarized in three broad phases: 

1) Calculation of the stiffness matrix, K, the mass matrix, M, the damping matrix, 
C, and the external load vector, R. 

2) Solution of the equilibrium equation, Equation (27). 

MU + CU + KU = R (27) 

U Vector of Nodal Point Displacement 

U Vector of Nodal Point Velocity 

U Vector of Nodal Point Acceleration 

3) Evaluation of the element stresses and resulting energy release rate. 

In this investigation, the transient dynamic analysis employed the Newmark 
Method of implicit time integration [32]. Since this method is unconditionally stable, then 
the interval of time advancement, time step, may be selected solely on the accuracy 
desired. Wave propagation problems are limited by characteristic times by which 
information can be physically communicated. The maximum time step is related to the 
acoustic wave speed and element size as described by Equation (28) and known as the 
Courant condition. 



46 



At 



max 



2c, 



(28) 



where L is the element length in the direction of the wave and c is the longitudinal wave 
velocity in an unbounded sample of the material. 
Calculation of the Energy Release Rate 

The use of effective elastic fracture theory offers the significant luxury of only 
having to conduct linear elastic analysis. The virtual crack extension method is used in 
ADINA to calculate the generalized expression of the energy release rate including the 
effects of thermal loads, pressure loads on the crack faces, and inertia forces. This method 
calculates the variation in total potential energy from a virtual displacement of nodes 
bounding the crack, represented in Figure 20. The energy release rate is defined by (Irwin 
[33] in 1956) as the ratio of the change in total potential energy to the change in crack 
area. The integral of Equation (29) is evaluated within the deformed region of the virtual 
shift. Note that the inertial energy is accounted for in the last term of Equation (29). 



I 



.■■..:- : :,.::,,--;':,:- : :- : :''':-:'1 



>: 



f 



a) Physical Problem 




Zone I, rigid body region shifted 
Zone II, distorted elemens 
Zone HI, unchanged by the shift 



b) Virtual Crack Extension 



Figure 20. Virtual Crack Extension Method 



47 



dAX 



du, 



fa 



4,J[ V y Sx t *' &, J 'd Xj *J Aj/dxj ' 



V 

s 

A c 
& 

fi 

t, 



°ij 



£ ij 



1 r .. du i 



*t\^^^^ 



■*e V 



dx.. 



Volume of the Cracked Body Enclosed by the Virtual Shift 

Surface of the Cracked Body Enclosed by the Virtual Shift 

Components of Virtual Crack Extension Vector 

Increase in Crack Area 

Kronecker Delta 

Components of the Body Force Vector 

Components of the Surface Traction Vector 



W JT* <Tyds t j = Total Stress Work Density 



Stress Components Acting on the Crack Volume 



Small Strain Components of the Crack Volume 



Mass Density 
Components of Acceleration 



(29) 



The energy release rate, G, was evaluated by applying Equation (29) along discrete 
paths defined by the "ring" of nodes shown in Figure 21. Selection of the virtual shift ring 
dictates the zone of distorted elements that contribute to the change in total potential and 
is therefore directly related to the accuracy of the calculation. Several virtual shifts were 
specified in the ADINA input file for establishing accuracy criteria of element sizes. 



48 






Multiple shifts defined 
for verifying convergence 



Half-length of crack, a 



Ring of Nodes for 
Virtual Crack Extension 




Figure 21 . Virtual Crack Extension Rings 



Setup of the Dynamic Analysis 

The numerical model was set up as a 2-D plane strain problem with symmetry 
about the plane containing the specimen notch and the axis of the SHPB. Further 
symmetry was prevented because of time-dependence on the evolution of the state of 
stress. One-half of the specimen was required to allow the stress wave to enter the 
specimen from the incident bar and exit to the transmitter bar. The bars were 
mathematically represented as extended lengths of the load distributors and thus the 
simulation did not have the actual change of diameter at the interface. They were included 
in the calculations to prevent artificial reflected stresses from free surfaces not existent in 
the actual experiment. The incident stress signals were modified through a force balance 
at the incident bar/load distributor interface where there was a change in cross-sectional 



49 

area. Auxiliary calculations proved this set up to be an acceptable approximation to the 
physical system. 

The loading functions (incident stresses) produced by the striker bar at various 
impact velocities were characterized in Appendix B for the actual SHPB. The simulations 
approximated the incident stress as a trapezoid with characteristics of Figure 22. The 
input file format of ADINA allowed for explicit description of the loading function. It was 
not feasible to model all ten specimen groups for all possible loading profiles. Therefore, a 
single profile, near the median of all loading profiles of Appendix B, was selected as a 
representative baseline. The incident stress was modeled by a 50 us rise time up to a 
magnitude corresponding to a 95 MPa stress amplitude in the incident bar. The stress 
pulse had a 250 u,s duration and was followed by a 75 us unloading time. 




50 us Rise Time 



100 




Timers 



— i — i — | — i — i — r- 



150 



200 



250 



300 



-i 1 1 1 r- 



350 



400 



250 (is Duration 



Figure 22. Modeling the Incident Stress Signal 



50 

Mesh and Element Selection 

Selection of element size was bounded by minimum time constraints for punctual 
calculations, as dictated by the Courant condition, and maximum element size suitable for 
representing the notch tip stress field gradients. A discrete crack approach was used for 
the notch because of known location and direction of crack propagation. Elements about 
the notch tips were represented by 8-node quadrilateral isoparametric elements having one 
side collapsed and nodes shifted to the quarter-point. These notch tip elements result in 
triangular elements that adequately capture the stress singularity. All other elements of the 
specimen and bar employed 8-node isoparametric quadrilaterals. The total number of 
elements varied according to specimen size and type of holed-notched cavity. Typically, 
the problems involved about 850 elements in the specimen and 250 elements in each bar. 

Element size and mesh configuration was validated through a series of calculations 
to investigate convergence of the energy release rate parameter. Analysis were made on 
specimens having a region with a fixed radius of 5 mm about the crack tip, as in Figure 21, 
that was subdivided into 2, 5, 9, and then 20 radial elements. The change in subdivision 
produced a proportional change in element size. Results of Table 6 show suitable 
convergence of G for element size of 1 .0 mm or smaller. The convergence study revealed 
that the virtual shift should be evaluated with an area encompassed by at least four element 
rings and having a maximum element size of 1.0 mm. 

The finite element mesh shown in Figure 23 represented the physical system of the 
experiment. The discontinuity of the notched cavity was represented through boundary 
conditions assigned to nodes on the plane of symmetry. The nodes of the notch were 
allowed to displace in the Y-Z plane, while the centerline nodes just outside the specimen 



51 



PHYSICAL SYSTEM 



Striker bar 




L.ft 4J. J 



ELEMENT REPRESENTATION — — 
Incident 





AD1NA MESH OF SPECIMEN 



mmk 




Figure 23. Finite Element Mesh of the Dynamic Experiment 



notch and in the incident and transmitter bar were kinematically restrained to only 
displacement in the direction of the applied load, Y-direction. 
Stability and Sensitivity Calculations 

Several design studies were completed to investigate the interplay between 
specimen dimension and input load function on the history of the energy release rate. 
All results will be presented as normalized energy release rate quantities using the G 
determined by static application of the peak load. The first study involved the effect of 
rise time on the inertial energy term in Equation (29) of the energy release rate. The 



52 



Table 6. Mesh Size Study for Energy Release Rate Convergence 
(using 102 mm cylinder with a/R = 0.25, r/R = 0.08 as the test case) 



Distance 


*G 


Distance 


*G 


Distance 


*G 


Distance 


'G 


R, mm 


2.5 mm 


R, mm 


1.0 mm 


R, mm 


0.56 mm 


R, mm 


0.25 mm 




Elements 




Elements 




Elements 




Elements 


0.0 


54.0549 


0.0 


54.0664 


0.0 


54.0694 


0.00 


54.0716 


2.5 


54.0477 


1.0 


54.0619 


0.556 


54.0655 


0.25 


54.0680 


5.0 


54.0801 


2.0 


54.0922 


1.111 


54.0958 


0.50 


54.0982 






3.0 


54.0951 


1.667 


54.0987 


0.75 


54.1011 






4.0 


54.0956 


2.222 


54.0992 


1.00 


54.1016 






5.0 


54.0953 


2.778 


54.0994 


1.25 


54.1018 










3.333 


54.0994 


1.50 


54.1019 










3.889 


54.0995 


1.75 


54.1019 










4.444 


54.0995 


2.00 


54.1019 










5.0 


54.0995 


2.25 
2.5-5.0 


54.1019 
54.1020 

■ 



*G in the table is the energy release rate in units of N/m per unit length of specimen. 



baseline rise time of 50 u,s was used, as well as, hypothetical rise times of 5 u,s and 100 us. 
The specimen for all graphs of Figure 24 had a 76 mm diameter and a cavity design with 
a</R = 0.45 and r/R = 0.12. Results in Figure 24 show significant inertial energy with the 
short rise time. The normalized energy release rate of the short rise time (5 u,s) becomes 
approximately 2.5 during initial loading of the specimen. This study demonstrates the 
synergy between time scales and its consequence on inertial effects. The 50 u,s rise time of 
the actual bar, as well as the 100 u,s rise time, show an insignificant inertial energy term; 
although, they do show a time-dependent attribute of the energy release rate. This 
transient response is a direct coupling of the loading function and specimen size. The 



53 



Rise Time of Incident Signal 

5-us Rise Time 

— +— 50-us Rise Time 

100-us Rise Time 




i i i i I i i i i I i — i — i — r 



50 100 150 200 250 

Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 

Figure 24. Effect of Rise Time on the Dynamic Energy Release Rate 



duration of the rise time of the incident wave and this transit time across the specimen act 
together to define the number of internal reflections of the stress wave within the specimen 
and its resultant notch tip stress field. It calls attention to possible limitations of the 
design space for applying the stress intensity functions obtained by Yang et al. [18]. 

The rise time study exposed the possibility that the shortest and longest rise times 
of the actual SHPB incident signals could present problems. An investigation was made to 
examine the various loading profiles produced by the SHPB, listed in Appendix B. The 
normalized energy release rate histories, given in Figure 25, show no concerns with the 
inertial term, but do show the time-dependent response seen in the previous study of the 
rise time. The time it takes for the normalized release rate to reach unity, call it t G , is a 
parameter to investigate and compare to the rise time of the transmitted signal, call it / T . 



1.20 





1.00 - 


So 


- 


»■£ 


0.80 - 


<D CD 




a: co 


_ 


>>s^" 




|o 


0.60- 


c o 




w E 




<1> c 


0.40- 


IM >■. 




is 9. 

E 

O 


- 


0.20 - 



0.00 



54 

Loading Profile from Varying Strfcer Impact Vekxiv 

— 3330 MPa Incident Stress with 80 us Rise Time 

1 50.58 MPa Incident Stress with 50 us Rise Time 




I I | I I I I [ M II | I I I I | II I I | I I II | I I I I II I I I | I I I I I I I || I I I II I 



10 20 30 40 50 60 70 80 90 100 110 120 
Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 



Figure 25. Response for Bounds of the Actual SHPB Loading Capability 



Equivalency of t G and t T implies that the stress intensity associated with the peak splitting 
tensile stress in the specimen is the same whether the peak was achieved statically or 
dynamically. The ability to directly use the stress intensity functions, F(a), of Yang et al. 
[18] for the fracture analysis requires the stress distribution to be equivalent to the 
quasistatic case. 

The energy release rate characterizes the stress distribution through its relationship 
with the stress intensity factor and, thus, can be used to validate the stress intensity 
functions in the dynamic loading cases. Therefore, the peak stress achieved at some time, 
t T , and the time it takes for the normalized energy release rate to reach unity, t G , should be 
of the same duration or else the specimen will fail at an uncharacterized stress intensity. 
Plots similar to Figure 25 were established for all loading scenarios produced by the split 
Hopkinson bar to establish t G values that were compared to the rise times, t T , of the 



55 



individual transmitted stress signals. The comparisons show that the times were equivalent, 
for all holed-notched specimen types subjected to the range of loading functions produced 
by the split Hopkinson pressure bar. 

An additional study was performed to verify that the appropriate time step 
requirements of the Courant condition were met. Results of calculations using five 
different time steps are presented in Figure 26, including two with time steps much less 
than the 100 ns step dictated by Courant condition . All five calculations used the same 
mesh with maximum element size near the crack tip of 1 mm. Results show that time 
steps at or below the Courant condition converge to the same value, as expected. Results 
also show time steps up to 1 us could be used without significant error, but time steps of 5 
us or greater begin to introduce error and are inadequate for use in this investigation. 
This study validated the use of the 100 ns time step for use with all further calculations. 



0.15 



T.O 0.10 

£1 

w J2- 0.05 

'*: E 
= i- 

£1 



0.00 



;=, -0.05 - 



-0.10 



Tine step used 
| — 10-ns time step 
' — 20-ns time step 
— 100-ns time step 

- 1 -us time step 



— 5-us time step 




t r 



i i r 



"i 1 1 r 



i 1 1 r 



"i r 



160 



180 200 

Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 



220 



Figure 26. Time Step Analysis for Solution Accuracy 






56 

Numerical Results of the Fracture Experiments 

The investigations of time step and rise time provided confidence in the numerical 
stability and validity of the proposed calculational set up so that detailed analysis of the 
holed-notched cylinders of the experiment could be conducted. Energy release rates were 
investigated for both static and dynamic loading of the specimens shown in Table 7. This 
representative set encompassed the broad range of cavity types, as well as specimen sizes. 

The holed-notched cylinder contains two notch tips and each must be considered. 
Quasistatic loading allows stress equillibrium to present identical stress intensity at each 
tip. However, dynamic stress wave loading presents asymmetric loading beginning at the 
incident bar side of the specimen, progressing along the cylinder diameter to the notch tip 
on the transmitter bar side, and then multiple internal reflections within the specimen. 
There was concern whether or not both notch tips exhibited the same evolution and 
amplitude of the stress intensity (interpreted via the energy release rate). The calculations 
for the specimens of Table 7 include results for both notch tips as functions of time to 
verify that symmetry requirements were met. 

The basic set up for the series of calculations are summarized for clarity. All 
calculations used a maximum element size of 1 mm about the notch tip, 0. 1 u,s time steps, 
virtual displacements at a distance of four element rings away from the tip, input loading 
function with 50 |is rise time up to a plateau stress of 150 MPa lasting for 250 [is, and all 
load distributors on the specimen had dimension 2t/D. Plane strain conditions with elastic, 
isotropic material constitutive equations were prescribed (and allowed as per the effective 
elastic crack theory). All results were normalized by the energy release rate obtained by 
quasistatic application of the same 150 MPa stress and times were all referenced to the 



57 

time at which the incident wave first reached the interface of the incident bar load 
distributor and specimen. 





Table 7 


Matrix of Finite Element Calculations 








76 


mm (3 inch) 


102 mm (4 inch) 


152 mm (6 inch) 








Diameter 


Diameter 


Diameter 


a/R = 0.25, 


r/R=0.0S 




X 


X 


X 


a/R = 0.45, 


r/R-0.08 






X 




a/R = 0.25, 


r/R=0A2 










a/R - 0.45, 


r/R=0.\2 




X 


X 


X 



The first two calculations were for cavity designs a/R = 0.25, r/R = 0.08 and a/R = 
0.45, r/R=0. 12 of the 76 mm diameter cylinder. Results in Figure 27 and 28 show that 
the normalized energy release rate quantity of both notch tips respond symmetrically and 
both reach a maximum of unity - - meaning, that at that instant, the stress intensity for the 
dynamic loading is equivalent to static application of the same peak load. Variations past 
the first peak are not significant for those specimens failing concurrently with the peak. 
Comparison of the time required for the normalized energy release rate, t G , and the 
observed time of the peak stress to develop, t T , (taken from the synchronized stress and 
fracture photograph histories in Appendix D) confirm that the specimen fails at the time 
domain of first peak in the normalized energy release rate. The time axis of the 
calculations, the stress signals, and the photographs are now all synchronized to the same 
coordinate system. Note that there is a time lag between the two notch tips because of a 
finite time of travel as dictated by the elastic wave speed and the distance between them. 




58 



1.20 




50 75 100 125 150 

Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 



Figure 27. Normalized G for 76 mm Diameter Cylinder with Cavity Design of a/R = 0.25 

and r/R = 0.08 



1.20 



o 

Z 




Holed-Notched Cylinder. 7 6 mm Diameter 
(a/r= 0.45. r/R = 0.12) 



First cracktip encountered 
Second cracktip encountered 



i i i | i i i i i i i i i 
25 



i | i i i i | i i i i | i i i i | 

50 75 100 125 150 175 200 

Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 



Figure 28. Normalized G for 76 mm Diameter Cylinder with Cavity Design of a/R = 0.45 

and r/R = 0.12 



59 



A graphic, qualitative representation of the energy release rate data is found 
through the mode I, tensile opening stress. A band plot of tensile stress is given in Figure 
29 for the 76 mm diameter cylinder with r/R = 0. 12 and aJR = 0.45 under quasistatic 
loading. Only one-half of the cylinder was modelled, but the band plot is given for the 
entire diameter. Plots from dynamic loading are given in Figure 30 for selected times of 
the initial loading event with time referenced to the same basis as the experiment. The 
sequence of mode I stress band plots can be compared to the energy release rate data of 
Figure 28 for the same type of specimen. The plots show that the notch tip stress fields 
are relatively symmetric past 50 us and that both appear to reach the static case. 




Figure 29. Mode I Stress Field for Quasistatic Loading of 76 mm Diameter Cylinder with 

Cavity Design of aJR = 0.45 and r/R = 0. 12 



60 




cs 

o 
II 

§ 

Tt 
o 

II 

1 

& 

•a 

">> 



is 
1 

s 

• »-« 

Q 

S 

E 

>o 
t-» 

tw 
o 

J 

«J 
O 

o 



cS 
T3 

a) 

CA 

en 

0) 

£ 

I— ( 

<u 
•a 
o 



o 

CO 
Vi 

E 



61 



Analysis of the 102 mm cylinders show the normalized energy release rate is 
heavilly influenced by wave interactions. Results in Figure 26 of a specimen with cavity 
design of r/R = 0.08 and aJR = 0.25, show both notch tips satisfy the quasistatic stress 
intensity conditions at the first peak. Results from a specimen with cavity of r/R = 0.12 
and aJR - 0.45, Figure 32, reveal that the energy release rate at first peak becomes 
approximately 1 10% of the quasistatic case. However, the stress intensity depends on the 
square root of the energy release rate as described earlier in Equation 28 and, thus, the 
dynamic stress intensity ratio is found by Equation 30 to be only about 5% greater. The 
error of order 5% was considered small enough to assume the 102 mm diameter holed- 
notched cylinders and their stress intensity functions, F(a), as valid fracture specimens. 

■is- dynamic "V Czc< 

Kf atic 4ge 



UlS I .or dynamic 

-=^ = .P— — -VTlO-1.05 (30) 

IGE V G stauc K ' 



static 





1.40 


m 




•4-» 

CO 


1.20 


a: 




8!o 

CO „ 


1.00 






a> m 






0.80 


d)rr- 




c CD 




C O 

Ulf 


0.60 


"O (D 




o> c 




N >, 

is. 


0.40 


E 




o 


0.20 



0.00 




Holed-Notched Cylinder. 102 mm Diameter 
(a/r = 0.25. r/R ■ 0.08) 

First crack tip encountered 

Second crack tip encountered 



25 50 75 100 125 150 

Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 



175 



200 



Figure 3 1 . Normalized G for 102 mm Diameter Cylinder with Cavity Design of o/R = 

0.25 and r/R - 0.08 



62 



0) 

a. 
jf 

*£ 

E>o 

?o 

LUg 
32 

E 
c 
o 

2 



1.20 
1.00 - 
0.80 
0.60 - 
0.40 - 
0.20 - 
0.00 




T — I I i I i — i — i — i — i — i — i — i — r 



Holed-Notched Cylinder. 102 mm Diameter 
(a/r = 0.45. r/R = 0.12) 



First crack tip encountered 
Second crack tip encountered 



i — i — r 



50 75 100 

Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 



125 



150 



Figure 32. Normalized G for 76 mm Diameter Cylinder with Cavity Design of o/R = 0.45 

and r/R = 0.12 



Analysis of the 152 mm cylinders revealed an increased problem with the transient 
response of the normalized energy release rate. The response, Figure 33 and Figure 34, 
show that the dynamic G exceeds the static value by approximately 38% for the cavity 
type ot'r/R = 0.08 and aJR = 0.25 and by nearly 60% for the cavity type of r/R = 0. 12 and 
aJR = 0.45. Applying similar analysis of Equation 29 for these two cases confer that the 
error in the stress intensity function is still an unsatisfactory 17% and 26%, respectively. 
The consequence of this analysis is that the 152 mm diameter specimens will not be used 
in the analysis of the fracture properties. To utilize these specimens one must obtain the 
time-dependent stress intensity functions and utilize an experimental procedure that 
precisely determines the time-of-failure. 

The oscillatory response of the normalized energy release rates compelled further 
investigation to see if it resulted from inertial effects, reflected wave superpositioning, or 



63 

both. The ADINA code allows explicit output of just the inertial term of Equation 29. 
Results of the inertial terms from one cavity type of each specimen size are shown in 
Figure 35 as normalized by their respective quasistatic energy release rate. The inertial 
term was found to reach only approximately 3% of the total energy release rate and, 
therefore, was not considered significant in any of the calculations. Elimination of inertial 
effects leaves the oscillatory nature of the normalized G curves dependent on wave 
reflections within the specimen. The graphs of Figures 27, 28, 3 1-34 show that the 
transient waves are related to specimen size (recall that the input stress function was a 
constant.) As the specimens become larger, the input loading function and reflected 
waves superpose to increase the stress intensity about the notch tips. This insight provides 
a basis for future efforts to modify the incident stress profile applied to a specific specimen 
geometry so that inertial and wave reflections are reduced. 



cc 



1.40 



1.20 - 



JX 0.80 



CO ., 

ijf 1.00 

CD CD 
*£ 

o y 

c o 

w E 0.60 - 

II 0.40 



o 

z 



0.20 



0.00 



Holed-Notched Cylinder. 152 mm Diameter 
(a/r = 0.25. r/R = 0.08) 



First crack tip encountered 
Second crack tip encountered 



TT 




"i i i i — | — I — I — I — i — | 

25 50 75 100 125 150 
Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 



Figure 33. Normalized G for 152 mm Diameter, a/R = 0.25 and r/R = 0.08 Cavity 



64 



I 

a? w 

c o 

T3 CD 
CD C 
N >> 



o 

z 



1.60 

1.40 

1.20 

1.00 

0.80 

0.60 - 

0.40 



0.20 



0.00 



Holed-Notched Cylinder. 152 mm Diameter 
(a/r = 0.45. r/R = 0.12) 



First crack tip encountered 
Second crack tip encountered 




H l"i I I I I I I I I I I I I l I I I i i i i i i i i i i i i i i i i 



25 50 75 100 125 150 

Time from Incident Wave Reaching the 
Bar-Specimen Interface, (us) 



175 



200 



Figure 34. Normalized G for 152 mm Diameter, a/R = 0.45 and r/R = 0.12 Cavity 



O 

hg, 

]§?? 
t: E 

CD CD 

SI- 
'S £: 

h CD 

o 55 
z't= 

CD 

B 



Cylinder Size and Cavity Type 
76 mm Diameter, a/r = 0.25, r/R = 0.08 
102 mm, a/r = 0.45, r/R = 0.12 
152 mm, a/r = 0.45, r/R = 0.12 




-0.03 



(All calculations used the same loading ramp rate and amplitude) 

1 M ' | M I I | M I I | I I I I | | | | | | | | | | | | | | | | | | | , | 

25 50 75 100 125 150 175 200 

Time from Incident Wave Reaching the 

Bar-Specimen Interface, (us) 

Figure 35. Normalized Inertial Term for Various Cylinder Sizes 






65 

Summary of the Finite Element Analysis 

Detailed finite element analysis provided insight to the dynamic fracture event not 
available through experimentation alone. A key observation was the time-phasing of the 
energy release rate and the incident stress signal during the dynamic event. Effects of the 
inertial energy term were shown to be tied directly to the time scales of the event. The 
coupling of the two times provide a method for validating the stress intensity functions 
used in this investigation, as well as, a method for future researchers to optimize the 
incident stress pulse commensurate with specimen dimensions. The energy release results 
were reported with the time basis referenced to same coordinate space of the 
experimentally obtained stress history and ultrahigh-speed photographs to provide a triad 
of information available for analysis. The three sets of data compliment each other to 
validate the time of failure and the state of stress at failure. 

The calculations were successful in validating the experimental technique for 
holed-notched cylinders of 76 mm and 102 mm diameter. On the other hand, the analysis 
showed that the 152 mm diameter cylinders could not be analyzed under the method 
proposed in this investigation. Results of the finite element analysis and the experimental 
effort are now carried forward to the nonlinear fracture models to extract the dynamic 
fracture properties. 






CHAPTER 6 
EVALUATION OF FRACTURE PARAMETERS 

The finite element analysis provided a method of interpreting the stress data for 
the 76 mm and 102 mm diameter cylinders. Application of the nonlinear fracture models 
was the final step in extracting the material properties and understanding the fracture 
response to dynamic loading. The experiment strength data were curve fit to allow direct 
implementation into the Size Effect Model (SEM) and Two Parameter Fracture Model 
(TPFM). Both models were programmed within spreadsheet software with automated 
routines to conveniently extract the fracture properties. 

The experimental effort provided a set of eight distinct specimen types to carry 
forward into the fracture analysis, Groups 1 through 8 of Table 5. Solution confidence 
was increased by using a sample basis with eight groups instead of the three groups 
called for in the discussions of Chapter 2 and the original derivation by Tang et al. [13]. 

Fracture Parameters using the Two Parameter Fracture Model 

Application of the Two Parameter Fracture Model 

Application of the TPFM was facilitated by programming the model into 
Microsoft Excel spreadsheets. The fracture mechanics equations and geometry 
dependent relationships presented back in Chapter 2 were entered directly within the 
spreadsheets along with cylinder geometry, strength response, and material properties. 
The spreadsheets were used interactively to provide initial acceptance of the raw 



66 



67 

experiment data, then allow access by curve fit and regression software, and finally 
provide solution of the fracture properties. 

The peak load version of the TPFM can be broken down into six primary steps of 
application. These steps are presented along with excerpts of the spreadsheet algorithms 
for a clear and documented presentation of the methodology. 

Step 1. The failure stress quantities from the experiment covered a strain rate 
regime of approximately 2 /s to 7 /s, in addition to the quasistatic rate. The regression fit 
of each group's stress response provided interpolated data at any desired strain rate. A 
common set of discrete strain rates was selected for applying the TPFM. The quasistatic 
tests and selected dynamic strain rates from 2 /s through 7 /s, with an arbitrarily chosen 
strain rate increment of 1 /s, were used to provide seven applications of the TPFM. 

The first step was to calculate K Ic and CTOD c using Equation (1) for a series of 
assumed values of the critical effective crack lengths a c . The nonlinear response of the 
concrete's fracture process zone creates an effective crack length at the failure state that 
is greater than the initial, physically cut crack length. In comparison, ideally brittle 
LEFM response would simply use the initial crack length. The range of values for a c 
began with the initial precut notch length and extended up to a maximum of 80% of the 
radius (the zone of mode I tensile stress for the splitting tension cylinder). Spreadsheets 
were established to calculate the K, c and CTOD c pairs. An example of the spreadsheets 
for all eight groups at strain rate of 2 /s is provided in Appendix E with detailed 
explanation. Each sheet required the following data to be input: material properties of 
elastic modulus and Poisson's ratio, peak stress (from regression equations of the stress 
data), and physical dimensions of the holed-notched cavity and cylinder geometry. 



68 

Each specimen type, or group, had different holed-notched cavity design and peak 
stress. These differences resulted in pairs of Ki c and CTOD c that were not of equal 
abscissa increments. The set of K Ic and CTOD c data for each group was then set to be 
curve fit to establish a means to evaluate each specimen group at the same K/ c increment. 

Step 2. Grapher® software extracted the K lc and CTOD c pairs from each group's 
spreadsheet to graph and curve fit them to a polynomial function. A third order 
polynomial was used to fit the holed-notched designs of r/R = 0.08, while a fourth order 
polynomial was used to fit the designs of r/R = 0. 12. The high order polynomials were 
used to ensure correlation coefficients, r 2 , of at least 0.998 or greater. 

Step 3. The curve fit equations of the K lc and CTOD c data were then supplied to a 
spreadsheet to recalculate CTOD c for each group at the same abscissa, K !c , increments. 
Then, the average CTOD c for all groups at each increment was found. Excerpts of the 
spreadsheets used for each strain rate with detailed explanation of the cells are provided 
in Appendix F. Care was taken to prevent erroneous extrapolation by only replicating the 
range of K Ic and CTOD c data originally produced by the experiment in the initial TPFM 
calculations of Step 1. This range was restricted to a subset created by the intersection 
operator performed on the intervals from all eight groups. 

Examples of these K Ic and CTOD c curves are provided in Figures 36-38 for the 
the quasistatic data. The graphs were segregated according to specimen diameter in order 
to maintain clarity. The plots of Figure 36 contain data of Group 1 through Group 4 (76 
mm diameter cylinders), Figure 37 shows Group 5 through Group 8 (102 mm cylinders), 
and Figure 38 contains data of Group 9 and 10 (152 mm diameter cylinders). Note, the 
152 mm data was only analyzed for the quasistatic loading rate. 



69 



0.020 



0.015 - 
? 

E 

jT 0.010 

Q 

p 

O 

0.005 - 



0.000 



Group # and Specimen Type 

Group 1, a/R=0.08, r/R=0.25 

# — Group 2, a/R=0.08, r/R=0.45 
-& — Group 3, a/R=0.12, r/R=0.25 
— | Group 4, a/R=0.12, r/R=0.45 




0.60 



0.80 



1.00 1.20 

K1c(MPa-m A 0.5) 



1.40 



1.60 



Figure 36. Curve Fit of CTOD c Versus K Ic for 76 mm Diameter Cylinders (Groups 1-4), 

Quasistatic Rate 



0.030 - 
0.025 



g" 0.020 

E 

o" 0.015 
Q 

O 



b 0.010 - 



Group # and Specimen Type 

Group 5, a/R=0.08, r/R=0.25 

-Q — Group 6, a/R=0.08, r/R=0.45 
-♦ — Group 7, a/R=0.12, r/R=0.25 
^K — Group 8, a/R=0.12, r/R=0.45 




0.60 



0.80 1.00 1.20 1.40 

K1c(MPa-m A 0.5) 



1.60 



1.80 



Figure 37. Curve Fit of CTOD c Versus K lc for 102 mm Diameter Cylinders (Groups 5-8), 

Quasistatic Rate 



0.035 

0.030 - 

^. 0.025 

E 

j= 0.020 

g 

§ 0.015 

° 0.010 

0.005 

0.000 



70 

Group # and Specimen Type 

— Group 9, a/R=0.08, r/R=0.25 
|— Group 10. a/R=0.08, r/R=0.45 




Quasi- static 



~i r i r | i : i i i — i — i — i — i — i — i — r 
0.80 1.00 1.20 1.40 



I ' ' ' ' I 
1.60 1.80 



K1c(MPa-m A 0.5) 



Figure 38. Curve Fit of CTOD c Versus K Ic for 102 mm Diameter Cylinders (Groups 5-8), 

Quasistatic Rate 



Step 4. The fourth step was to calculate the least squared error for each possible 
increment of effective crack length, ct c , using Equation (6), but restated as Equation (3 1) 
in an updated form. This step was conducted within the same spreadsheets of Step 3 and 
found in Appendix F. 



Minimized Error ■ £ \cTOD c )"* - (CTOD c )' ] 

;=1 



(31) 



Existence of a minimum in the open interval of {K Ic min , Ki c max ) was not guaranteed 
to exist nor be unique. However, the data of this investigation did produce a global 
minimum for each strain rate and, in fact, the minimums were inclusive of the closed 
interval [K Ic min , K^] for all strain rates. 

The quasistatic data in Figures 36-38 were analyzed in two methods: one using all 
three diameters and one using only the 76 mm and the 102 mm diameters. The dynamic 



71 

data analysis included only the 76 mm and 102 mm diameter cylinders. The least squares 
difference of CTOD for all ten groups is given in Figure 39, while the CTOD differencing 
of just the 76 mm and 102 mm diameter cylinders is given in Figure 40. Two minimums 
appear in Figure 40 to allude to the fact that minimums are not guaranteed to be unique, 
but the global minimum is taken as the apparent fracture property. 

Step 5. The mutual solution to the set of all specimen groups was found by 
locating the effective crack length that corresponded to the minimum error. The fracture 
parameters K Ic and CTOD c at this minimum error were taken to be material properties 
and given the designation of Ku and CTOD*. The critical effective notch lengths, a c , 
were also extracted and recorded for each respective strain rate. 



D.UC-UO 






jT 5.0E-05 : 

E 

£- 4.0E-05 ■ 




/ 


| 

ul 3.0E-05 ; 






:ed 






•| 2.0E-05 : 

c 

i 1.0E-05 : 






0.0E+00 : 


I 1 


'1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 



0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 

K1c (MPa-m A 0.5) 



Figure 39. Minimized error for Quasistatic Data using 76 mm, 102 mm, and 152 mm 

Diameter Cylinders (all Groups 1-10) 






72 




•O.OE+00 



0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 

K1c(MPa-m A 0.5) 



Figure 40. Minimized Error for Quasistatic Data using only 76 mm and 102 mm 

Diameter Cylinders (Groups 1-8) 



Note that the nomenclature used in the initial discussion of the TPFM back in 
Chapter 2 resolved the fracture parameters as AT//* and CTODf* with the superscript 
signifying Shah's model [11]. The superscript, S, was dropped to prevent a proliferation 
of labels and designators that deter from the analysis. 

Step 6. The previous five steps were repeated for strain rates in the range of 2 /s 
through 7 /s at arbitrarily selected intervals of 1 /s. The spreadsheets established for each 
group at the 2 /s rate were used as templates for the repetitious calculations for all rates. 
Results of the Two Parameter Fracture Model 

Results from application of the TPFM to selected strain rates are presented in 
Figure 41 for K Ic * as a linear relationship with strain rate and in Figure 42 for K Ic ' as a 
logarithmic relationship with strain rate. The monotonic response of fracture toughness 
with strain rate is analogous to the peak stress data. The second characterizing fracture 
parameter, CTOD c \ is given in Figure 43. Both parameters are given for the quasistatic 



73 

rate and the strain rate interval from 2 /s to 7 /s. The Ki c * and CTOD c * data show no 
asymptotic response at the higher strain rates, thus, an indication that rate effects continue 
past the rate interval investigated. The results will be given later in tabular format with 
the results of the Size Effect Model. 



§ 300 

< 

E 

z 

9= 2.50 - 



o 

* 2.00 

w 

& 1.50 

■ 

W 

W 

6 1 -°°- 



8 



o 



0.50 




X 



Results from Two Parameter Fracture Model 
Linear Fit to the TPFM Results (Dynamic Only) 



1 '' i | ' i i i | i i i i | i i i i | i i i i | i i i i | i i i i i 

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 

Strain Rate (1/s) 



Figure 41. Critical Stress Intensity Results from the Two Parameter Fracture Model 



ci 

< 

E 
■ 

O. 
o 

S 

>i 

m 
w 

c 

0) 

1 

I 

CO 

8 

1 

o 



10.00 



1.00 



0.10 




X 



Results of Two Parameter Fracture Model 
Numerical Fit through the TPFM Results 



Ml | ' i ""»| i i niiii| i i iiiin| i 1 1 1 m rif i i iiiiiij — i i iiiinj — i 1 1 M i n i — i 1 1 H i m — | 1 1 ll m i 

1.0E-9 1.0E-8 1.0E-7 1.0E-6 1.0E-5 1.0E-4 1.0E-3 1.0E-2 1.0E-1 1.0E+0 1.0E+1 



Strain Rate (1/s) 



Figure 42. Strain Rate Trend of Critical Stress Intensity Factor 



E 
E 



8 
I 

Q 



0.03 



0.02- 



£ 0.01 

! 

o 

8 

| o.oo 



74 



X Results from Two Parameter Fracture Model 
Linear Fit to the TPFM Results (Dynamic Only) 




"i i i i 1 i i i i l i i i i I i i — i — i — I — i — i — i — i — I — i — i — i — i — i — i — i — i — r 



0.0 1.0 2.0 3.0 4.0 5.0 

Strain Rate (1/s) 



6.0 



7.0 



Figure 43. Critical Crack Tip Opening Displacement Results from the TPFM 

The theory behind the TPFM requires knowledge of both K/ c * and CTOD c * to adequately 
describe the fracture state. Therefore, the two are plotted against each other in Figure 44 
to provide a failure surface for the fracture states over all strain rates. The fit to the 
failure surface may serve to obtain estimates of strain rates in between the static and 
dynamic rates. The curve fit, Equation (32), was used to calculate quasistatic results 
from other research using a similar concrete mix (similar strength and elastic modulus) . 
Quasistatic fracture toughness results published by Tang et al. [13] for concrete with 
modulus, E = 26.5 GPa, were replicated within 13%. No published data were found to 
validate the equation to other dynamic results. 



K) c = 0.70565 + 20.955(C7DD; )+ 383 1 .&(cTOD* c f 
K Ic Critical Stress Intensity Corresponding to CTODc 

CTOD c Critical Crack Tip Opening Displacement from the TPFM 



(32) 



75 



in 



1 

cb 

Q. 


3.00 - 


2.50 -_ 


o 

O 

a 

CD 

LL 


2.00 -_ 
1.50 - 


t 

CD 

c 


1.00 ^ 


85 
25 


0.50 -_ 


CO 

o 

IP 


0.00 - 






Quasi-static result 



* 



/0< 



y 



X 



X 



X Experiment data 
- — Curve fit, K1c= 0.70656 + 20.955*CTODc + 3831.8*(CTODc) A 2 



"i — r 



i — r — i — r 



"i i — i — r 



i — i — i — r 



i i i — r 



0.0000 



0.0050 0.0100 0.0150 0.0200 

Critical Crack Tip Opening Displacement (mm) 



0.025 



Figure 44. Failure Profile of the TPFM for all Strain Rates 



Fracture Parameters using the Size Effect Method 

Application of the Size Effect Method 

The SEM theory was also programmed into Excel® spreadsheets as analysis tools. 
The spreadsheet was designed to accept description of the cylinder geometry, stress 
versus strain rate data, and perform linear regression of the linearized SEM terms 
described in Equation (10). The SEM parameters of critical energy release rate, G f , and 
crack extension, c f , were also calculated within the spreadsheet. The steps required for 
applying the SEM model are given in a format compatible with those that described the 
TPFM. The SEM was applied to stress data from the same strain rate intervals used for 
the TPFM. 

Step 1. Geometry data of cylinder diameter, length, and notch cavity dimensions 
were entered and geometry functions of Equation (9), (16), and (17) were calculated. 



76 

The spreadsheets for all strain rates with detailed explanation are given in Appendix G. 
Failure stresses for the dynamic experiments were calculated using the linear curve fit 
equations of Table 5. 

Step 2. The quasistatic data was analyzed in two methods. The first considered 
all groups including the 1 52 mm diameter cylinders, while the second only used the 76 
mm and 102 mm diameter data. The linearized variables, Zand Y, of the SEM were 
calculated for each group using Equation (10). A separate spreadsheet was established 
for conducting these calculations at each strain rate interval. 

Step 3. The linearized variables were used to calculate the energy release rate, Gf, 
and critical effective crack extension, c/, as described by Equation (10). The energy 
release rate was recast into the stress intensity parameter, Kp, via Equation (12) and Cf 
was formulated into CTODf through Equation (13). The transformations were conducted 
to provide comparison of results between the two nonlinear fracture theories. 
Results of the Size Effect Method 

Results from the SEM and the TPFM are given in Table 8 in the nomenclature of 
each respective model and also with the SEM quantities recast to stress intensity and 
crack tip opening displacement for comparison with the TPFM. The stress intensity 
quantity, K$ from the SEM is plotted as a function of strain rate in Figure 45 along with 
the TPFM quantity, K tc . The two stress intensity parameters have similar response to 
strain rate and are within 1 1% of each other over the entire rate interval investigated. 
The difference between the two stress intensities may indicate a degree of nonlinear 
effects influencing the failure process. Quantities K lc * and £//were both obtained on a 
relatively limited range of specimen size but K\f is the stress intensity for a specimen of 



77 



Table 8. Discrete Value Results from Both Nonlinear Fracture Models 




Two Parameter 
Fracture Model 




Size Effect Method 






recast to TPFM quantities 


Strain 
Rate, 1/s 


K Ic CTOD c 
MPa-m 1/2 mm 


Gf 
N/m 


c f K If CTOD f 
mm MPa-m ,/2 mm 


10" 8 


0.86 0.00414 


24.67 


1.428 0.84 0.00352 


2 


1.71 0.01476 


92.90 


7.225 1.63 0.01541 


3 


1.94 0.01532 


115.82 


5.676 1.82 0.01526 


4 


2.15 0.01620 


141.26 


4.866 2.01 0.01568 


5 


2.40 0.01800 


172.32 


4.406 2.22 0.01644 


6 


2.69 0.02060 


206.46 


4.135 2.43 0.01742 


7 


2.93 0.02143 


243.69 


3.974 2.64 0.01855 



5 3.00 
o 

< 

E 

a. 2.50 



* 2.00 - 

o 

o 

5 1.50- 

CO 

I 1.00 



in 
tn 

a 050 

CO 




Q Results from Size Effect Method 

X Results from Two Parameter Fracture Model 



1 ' i i | i i i i | i i ii | i i i i | i i i i | i i i i I i i i i 

00 1.0 2.0 3.0 4.0 5.0 6.0 

Strain Rate (1/s) 



7.0 



Figure 45. Critical Stress Intensity Results from the SEM and TPFM 



78 

infinite size, thus, the notch is very small with respect to the specimen and LEFM theory 
applies. (By having the notch much smaller than the specimen the nonlinear effects also 
become much smaller to the global structure and LEFM theory is appropriate.) LEFM 
failure requires less energy than that of nonlinear fracture energy dissipation. Therefore, 
as a material approaches LEFM behavior the critical stress intensity from the two models 
become equivalent. The fact that the two models do not converge may be an indication 
of the degree of nonlinear fracture response that still exists at the higher strain rates. 

The SEM quantity c f was cast into the CTOD quantity of Equation (13) through 
the assumption that the initial crack length and increment of crack extension had the 
relationship of a »Cf. Results of Table 8 show that c f was indeed small with respect to 
the initial notch lengths. The comparison of the crack tip opening displacement from 
both models is given in Figure 46 and shows excellent agreement. The monotonic nature 
of their response is directly influenced by the linear regression of the peak stress data. 
The dynamic data again shows an uncoupled trend with the quasistatic data that will be 
acknowledged without comment because of the absence of data in that interval. 

The failure response describing the failure state by pairs of A//and CTODf were 
joined with the response plot of the TPFM to give the comparison of Figure 47. The 
failure responses are in good agreement and provide a degree of confidence in the failure 
results. The curve fit of the SEM response, included for continuity with the TPFM plot, 
shows an inversion between the quasistatic and dynamic data that likely indicates that a 
continuous description between the rate regimes is not appropriate. The failure response 
data at the higher strain rates show an initially steep slope and then transition to a fairly 
constant slope. The initial rise, corresponding to strain rates of 2 /s to 4/s, may suggest 



79 



E 

£ 0.025 

Q 
O 

P 0.020 



o 
o 
Q 
O 
h- 
O 

ci 
w 

Q 

e 
'c 

<D 
Q. 

O 

Q. 



o 

CD 

1— 

O 



0.015 



0.010 



0.005 - 



0.000 




Q Results from Size Effect Method 

X Results from Two Parameter Fracture Model 



"i i i i l i i i i I i i i i i i i i i — i — i — i — i — i — i — i — i — i — r 



0.0 1.0 2.0 3.0 4.0 

Strain Rate (1/s) 



5.0 



6.0 



7.0 



Figure 46. Critical Crack Tip Opening Displacement Results from SEM and TPFM 



o 
< 



o 

o 

CO 



in 

c 

0) 



(f> 



8 

•c 
O 



3.00 

2.50 - 

2.00 

1.50 

1.00 - 

0.50 

0.00 



X 



TPFM using experiment data 

Curve fit of TPFM, K1c = 0.70656 + 20.955*CTODc+ 3831.8*(CTODc) A 2 

SEM using experiment data 

Curve fit of SEM, K1f = 1.222 - 1 55.53'CTODf + 12708*(CTODf) A 2 



* 




/X 



Quasi-static results 



i — i — i — r 



~\ — i — i — r 



~i — i — r 



"i — i — r 



T 



t — i — i — r 



0.0000 0.0050 0.0100 0.0150 0.0200 

Critical Crack Tip Opening Displacement (mm) 



0.0250 



Figure 47. Failure Response of SEM and TPFM 



80 

a transition phase of material properties exiting the moderate strain rate regime. Recall, 
that the strength data was observed to have a transition region in the samel /s to 10 /s 
region. 

Relation to Linear Elastic Fracture Theory 

In general, as strain rate or loading rate increases the failure characteristics of a 
material are postulated to become more brittle and LEFM theory more applicable [3,4]. 
The fracture parameters from this effort were re-analyzed to extract information that 
describes whether the nonlinear effects are becoming greater with strain rate, remaining 
the same, or reducing to a more linear elastic description. One way to address that 
question was to investigate the growth of the fracture process zone ahead of the crack tip. 
If the nonlinear effects were to reduce with strain rate then the size of the zone would 
become smaller and the size of the effective crack length would tend to the size of the 
initial notch length. The critical effective notch length was extracted for each specimen 
group evaluated at Ki* for the strain rate increments from 2 /s to 7 /s and the quasistatic 
rate. The spreadsheets utilized earlier for the TPFM analysis were also used for this task. 
The increment of effective crack extension was found by Equation (33), then the average 
at the respective strain rate was calculated and given in Figure 48. The 76 mm and 102 
mm diameter data show a discontinuity between the quasistatic value and the dynamic 
data. No presumptions are made to describe this disconnect because no data was 
obtained between the 10" 8 /s and 2 /s interval. The dynamic crack extension length is 
larger than the static, meaning that the fracture process zone continued to develop from 
the 10~ 8 /s regime to the higher rates. However, the crack extension was observed to 
remain at a fairly constant value over the higher strain rates. 



81 



Ci 

a c 
a 



c =a —a 

"l M c **0 



Increment of Effective Crack Extension (mm) 
Critical Effective Crack Length (mm) 
Initial Notch Length (mm) 



(33) 



E 
E 



o 
'u> 

c 

0) 

X 

LU 

-*: 

o 

CO 

1— 

O 

o> 

.> 

i 

HI 

o 



14.00 - 

12.00 

10.00 

8.00 - 

6.00 

4.00 

2.00 



O 0.00 



% Quasi-static critical crack extension 
+ Dynamic critical crack extension 
— — Linear fit to the dynamic values 



i i i I i i i i I i i i i I i i i i I i i i i I i i i i l i i i i 



0.00 



1.00 2.00 3.00 4.00 5.00 

Strain Rate (1/s) 



6.00 



7.00 



Figure 48. Strain Rate Trend for the Critical Effective Crack Length 



Looking directly at the applicability of linear elastic fracture mechanics can assess 
the degree of nonlinear response. The LEFM value of critical stress intensity was 
evaluated for all the failure stress data and compared to the results of the nonlinear 
fracture theories. The K Ic from LEFM was found using Equation (34) with the critical 
crack length at unstable failure being the initial crack length. The form of Equation (34) 
immediately suggests that K Ic obtained by LEFM would be less than that of the SEM or 
TPFM solely because of its proportionality with the crack length. 



82 



K tc =o Nc ^m' F{a ) 



(34) 



recalling that oco is the initial crack length to specimen size ratio. 

The graph of Figure 49 shows the material response does not approach the LEFM 
solution and thus is still controlled by nonlinear effects. The abundance of data symbols 
in the LEFM analysis of Figure 49 was the result of each group producing its own 
fracture property in accordance with its initial crack length; while, the nonlinear analysis 
involved a single effective crack length for all groups. 



2 3.00 
o 

< 

E 

Q. 2.50 - 



* 2.00 - 

i_ 

o 
o 



1.50 



to 

I 1.00 

fi 

~ 0.50 

C/D 



H 







;r Fracture Model, all Groups 




Twu PaiaineU 


— 


— Size Effect Method, all 


Groups 


o 


LEFM, Group 1 


A 


LEFM, Group 5 


+ 


LEFM, Group 2 


X 


LEFM, Group 6 


o 


LEFM, Group 3 


* 


LEFM, Group 7 


D 


LEFM, Group 4 


• 


LEFM, Group 8 




1 ' ' ' I I ' ' ' I I I ' I I I I I I I I I I I I I I I I I I I I I I 

00 1.0 2.0 3.0 4.0 5.0 6.0 7.0 

Strain Rate (1/s) 



Figure 49. Comparison of Nonlinear Fracture Theory and LEFM 



83 
Summary on the Application and Results of the Fracture Models 

The peak load technique for both the Size Effect Method and Two Parameter Fracture 
Model produced fracture parameters with an approximately linear relationship with strain 
rate over the interval from 2 /s to 7 /s. Results are in agreement with the research of Oh 
[5] whose results of fracture energy as a function of loading rate also show a monotonic 
relationship; albeit, his results were for a lower dynamic regime. The concomitant 
application of the two models showed excellent agreement and provided independent 
verification of results. This investigation was the first documented research effort that 
applied both the SEM and TPFM models to the same set of data and the first to apply 
them to dynamic loading rates. 

Fracture parameters characterizing the size of the nonlinear fracture process zone 
ahead of the crack tip provided a metric of the divergence from linear elasticity. The 
critical crack extension, c/, provided insight to the size of the nonlinear fracture process 
zone and served as an indication that the fracture process zone still existed at the higher 
rates. Additional information from this investigation was the development of the failure 
response curve for both models in terms of stress intensity and crack tip opening 
displacement pairs that characterize the suite of failure states. 






CHAPTER 7 
SUMMARY AND CONCLUSIONS 

The objective of this research was to develop a method to obtain the dynamic 
fracture toughness of cementitious materials. Moderate strength concrete was selected as 
the material of study. The objective was achieved through the development of a system 
using the split Hopkinson pressure bar and specific holed-notched fracture cylinders that 
could be analyzed using a variant of two nonlinear fracture theories. The experiments and 
theory were complimented with a detailed finite element investigation that proved to be an 
integral piece of the research. 

Nonlinear Fracture Theories 

Two nonlinear fracture theories were implemented in a modified form that 
accommodated the data types available from the experiment apparatus. The innovative 
use of a peak load method provided a valid alternative to conventional diagnostic 
instruments, such as crack opening displacement gages, typically used in the (static) 
concrete fracture community; thus, circumventing their destruction in the dynamic 
environment of the split Hopkinson pressure bar. Both nonlinear models called for 
exploitation of size and/or geometry to apply the peak load method. One detriment of this 
method is the requirement of numerous specimens that must be tested. 

An understanding of the theories was used to specify fracture specimen size and 
geometry that minimized random and physical measurement errors for increased solution 



84 



85 

confidence. Ten distinct groups of fracture specimens were tested to satisfy the statistics 
of the peak load method. 

Fracture Experiments 

The primary piece of information required for this investigation was the peak stress 
at specimen failure over a range of loading rates. The split Hopkinson pressure bar 
proved to produce the information with a high degree of confidence because of a sound 
theoretical basis and a well-characterized experimental procedure. All stress histories 
were captured for detailed analysis and examined for anomalous behavior. 

A total of 33 quasistatic and 104 dynamic experiments were conducted to provide 
peak stress values of concrete fracture over a broad range of strain rates. The tests 
involved ten groups with distinctly different stress responses through differences in size, in 
geometry of the holed-notched cavity, or both size and hole design. The dynamic stress 
data produced linear response characteristics over the strain rate interval from 2 /s to 7 /s. 
Regression analysis of the peak stress data provided a critical link to carry the stress data 
into the fracture theories and evaluate all specimen groups at common strain rate intervals. 

The use of digital macrophotography provided a technique to synchronize the 
physical crack propagation event with the failure stress and finite element data. The 
photography captured evidence of symmetric crack propagation at both notch tips, pre- 
peak crack growth leading up to catastrophic failure at the peak stress, and single crack 
propagation from the notch tip. This evidence was essential to the validation of stress 
intensity functions and assumptions made on the analysis of the experimental procedure. 



86 

Finite Element Analysis 

Results of the finite element simulations were the final process in verifying the 
experimental technique. The calculations demonstrated equivalent stress intensities about 
both notch tips and provided insight to the temporal phasing between the loading history 
and elastic wave transmission time through the specimen. The synchronization of the 
experiment stress signal, crack propagation photographs, and finite element stress intensity 
calculations provide a triad of reinforcing analysis tools that validate the fracture results. 

Inertial effects were shown to not influence the energy partitioning process during 
mode I tensile loading and crack initiation. However, the calculations did reveal a 
significant inter-relationship between the loading profile of the incident stress and the 
specimen size. The stress profiles produced by the SHPB of this research prevented the 
use of established stress intensity functions for analyzing the 152 mm diameter specimens. 
The FEM analysis supported the use of 76 mm and 102 mm diameter cylinders but refuted 
the use of 1 52 mm cylinders. 

Behavior of the Fracture Response 

The fracture parameters of K lc and CTOD c were observed to monotonically 
increase over the strain rate interval from 2 /s to 7 /s. The dynamic values of critical stress 
intensity became as great as 3.4 times the quasistatic values. The strain rates achieved 
from the stress wave loading were significantly higher than that produced by any other 
published research. Results from both fracture theories were in excellent agreement at the 
quasistatic and over the dynamic strain rate range. Results were also consistent with 
observations of other researchers whom investigated fracture properties at low to mid 



87 

strain rates. A failure surface characterizing the K Ic and CTOD c quantities at failure over 
all strain rates were obtained for an improved understanding of the material response. 

Future Research 

Several additional opportunities for further study were recognized during the 
course of this work or were created by the work itself. These research ideas are presented 
for recommended areas of opportunity within the study of dynamic material properties and 
experimental techniques. 

The holed-notched cylinders are just one of a myriad of fracture specimens 
available for use in the split Hopkinson pressure bar. Additional specimen geometries and 
various notch configurations could present even more distinct groups. A criterion was 
presented for quantifying the level of distinction offered by various specimen designs and 
for increasing solution confidence. The holed-notched cylinders offered theoretical 
understanding of the stress distribution and alternate specimen designs should also be well 
understood. Suggestions of additional geometries include the eccentric compression 
specimen and the three-point notched beam specimen. Stress intensity functions exist for 
these two geometries and they are also used extensively in quasistatic testing [19]. 

The size and geometry of the holed-notched specimens were influenced by 
requirements of the concrete casting process and the necessity to maintain a minimum size 
ratio of the aggregate-to-specimen dimension. Therefore, the size design space can be 
extended and redefined for use with other materials. The nonlinear theories can be applied 
to high toughness steels, polymers, and even composites. The peak load method is not 



88 

material dependent and it presents an analysis tool that can be used for a broad class of 
fracture problems. 

An additional opportunity in specimen design involves the wave reflection 
phenomenon observed in the finite element simulations. One could design specimens that 
mitigate or reduce the superposition of internally reflected waves. Another approach to 
reducing the transient would be to investigate methods of shaping the incident stress so 
that the wave superposition within the specimen does not create oscillatory stress intensity 
response. It is desirable to get the specimen to respond with constant stress intensity over 
the time interval of crack propagation. Information is available for contouring the incident 
stress waves of the SHPB [34]. 

The integration of the macro photography with the split Hopkinson bar experiment 
proved a valuable addition to study dynamic fracture. The camera also provides a means 
to obtain crack velocity and time-to-failure data. This information presents opportunities 
to link material properties to crack velocity. 

Concluding Remarks 

An experimental procedure and accompanying theoretical analysis was developed 
to produce a well-characterized technique for quantifying dynamic fracture properties. 
The technique was based upon experimental, analytical, and finite element simulation 
efforts to calculate mode I fracture properties. Results confirmed preliminary conjecture 
that fracture properties of concrete increase with strain rate within the same interval that 
the tensile strength becomes rate sensitive. Concrete continued to exhibit nonlinear 
fracture response over strain rates from 2 /s to 7 /s. The nonlinear traits were documented 



89 

in terms of size dimensions of the fracture process zone and divergence from linear elastic 
fracture mechanics theory. 



APPENDIX A 
COEFFICIENTS FOR CRACK TIP OPENING DISPLACEMENT PROFILES 

The crack tip opening displacement profiles were found by Yang et al. [18] from 
elastic analysis using finite element methods. The profiles of Equations (18) and (19), 
Chapter 3, and are restated here for completeness. The common function, V(ci(/a), 
contains coefficients dependent on a/R and were given for discrete values of a/R in Yang 
et al. [18]. The discrete points were fit to polynomial forms that can now be applied to 
any increment of a/R, as required for use in the nonlinear fracture models. 




CTOD = COD(y=ao) 

1 



advancing 



nitial crack 



Figure A-l. Definition of Crack Tip Opening Displacement (CTOD) and Crack Opening 

Displacement (COD) 



Crac k Tip Opening Displacement Equations for Holed-Notched Cylinders with- 



r/R = .08 H(^,a) = F(--°-) 



-0.046 + 2.75a -3.28a 2 +4.6a 3 +0.7a 4 



a 



r/R = O.J 2 H&-,a) = V&-) 

a ' v a ' 



0.472 + 6.2a - 8.82a 2 + 10.86a 3 + 2.52a 4 
(1 + a)a 



90 



91 
where the common Vfag/a) function is expressed as, 



a„ 



a a 



where 

r Radius of Hole Cavity, m 

R Radius of the Specimen, m 

a Initial Notch Length, m 

a Current Notch Length, m 

a Ratio of a/R 



(n\ l fn \ 



«0 

\a J 



+L* 



a o_ 

\ a J 



3 



Coefficients I, were fit to polynomial form, L t =a + ba + ca 2 +da 3 + ec 4 + fa 5 and 
are given in Table A-l. 

Table A-l. Polynomial Coefficients to Crack Tip Opening Displacement Equations 



/ 



r/R = 0.08 

L0 5.70420 -26.7599 62.4935 -67.2030 27.3182 

Ll -10.79050 35.2868 -46.9876 22.7944 

L2 10.30400 -23.5758 23.0085 -9.3083 

L3 -4.09059 4.6976 -1.6452 

r/R -0.12 

L0 14.94900 -90.3581 232.1700 -266.8680 113.7160 

Ll -40.19580 248.7500 -643.1590 747.9760 -321.6440 

L2 76.26610 -673.5500 2590.2200-4960.6900 4650.9800 -1703.25 

L3 -34.79520 312.7250-1231.0200 2394.7500-2272.7000 840.79 



92 



Coefficients for CTOD of Holed-Notched r/R=0.08 



6.00 



~ 4.00 — 



-6.00 — 




I I I I 



I I ! 1 



I I I I 



I I I I 



I I I I 



0.20 0.30 0.40 0.50 0.60 0.70 0.80 

Ratio of Crack Length / Specimen Radius (a/R) 



Figure A-2. Coefficients for r/R=0.08 



• 


LOfrom Yang (18) 
L0, Polynomial Fit 




■ 


L1 from Yang [18] 


— - 


L1. Polynomial Fit 


A 


L2from Yang [18] 
L2, Polynomial Fit 




+ 


L3from Yang [18] 





L3, Polynomial Fit 



c 
<D 
D 

m 

CD 
O 

O 

0) 

o 

CD 



8.00 

6.00 - 

4.00 

2.00 - 

0.00 
-2.00 
-4.00 
-6.00 



-8.00 



Coefficients for CTOD of Holed-Notched r/R=0, 12 



» - A 



I I I I 




0.20 0.30 0.40 0.50 0.60 0.70 

■ Ratio of Crack Length / Specimen Radius (a/R) 



1 

0.80 



• 


LOfrom Yang [18] 
LCI, Polynomial Fit 




■ 


L1 from Yang [181 


— - 


L1, Polynomial Fit 


A 


L2 from Yang [18] 





L2. Polynomial Fit 


+ 


L3fram Yang [18] 





L3, Polynomial Fit 



Figure A-3 . Coefficients for r/R=0. 1 2 



APPENDIX B 
INCIDENT STRESS CHARACTERIZATION OF THE SPLIT HOPKTNSON 

PRESSURE BAR 

Characteristics of the incident stress wave were investigated by varying the striker 
bar velocity of impact. Gas pressure of the chamber used to accelerate the striker bar was 
selected and changed to vary the velocity. Results are given in Table B-l and show 
changes in the incident stress plateau as well as in the rise times. Definitions of the 
characterizing parameters are given in Figure B-l. 

Table B-l. Characterization of the Split Hopkinson Bar Incident Stress Wave 



Chamber Pressure, Rise Time of Stress Plateau Amplitude, MPa 



psi 



Incident Wave, us 



100 80 33.30 

150 65 68.05 

200 50 101.35 

300 50 150.58 

400 45 180.98 

500 45 209.94 



93 



94 



Amplitude of 
Stress Plateau 




-110 



~l — 1 — I — I — I — I — I — I — r- 



Time, ^s 



350 400 



— I 1 1 1 1~ 






5 





100 


150 


200 


250 


300 


4 


■w 


< 










w 




r 












> 



Rise Time 



Duration of Stress Plateau 



Figure B-l. Parameters for the Incident Stress Wave Characteristics 






APPENDIX C 
DYNAMIC STRENGTH RESULTS OF HOLED-NOTCHED CYLINDERS 



Table C-l. Dynamic Strength of Holed-Notched Cylinders, 76 mm (3 in) Diameter 



Strain Rate 


Group 1 


Group 2 


Group 3 


Group 4 




Peak Stress 


Time-to- 


Peak Stress 


Time-to- 


Peak Stress 


Time-to- 


Peak Stress 


Time-to- 


m/m/s 


MPa 


Failure, (is 


MPa 


Failure, (is 


MPa 


Failure, us 


MPa 


Failure, fis 


1.8 






4.089 


125 










1.9 














2.937 


120 


2.0 


5.778 


110 














2.1 






4.557 


113 


5.205 


120 






2.2 










4.323 


105 


3.523 


120 


2.3 


5.371 


87 


3.758 


117 










2.6 










5.833 


90 






2.7 






4.261 


100 






3.475 


100 


2.8 


7.488 


105 














3.0 






4.599 


78 






4.171 


90 


3.1 






4.482 


72 










3.3 


6.964 


88 






7.026 


85 






3.7 














4.895 


76 


3.9 


8.977 


90 


5.709 


85 










4.0 










8.191 


90 






4.1 


8.349 


86 


6.191 


75 






5.874 


82 


4.2 






5.074 


75 


6.798 


90 






43 


10.508 


70 














4.6 










7.391 


90 






4.7 


7.474 


74 










5.247 


90 


4.8 






5.950 


68 


8.873 


75 






5.0 


6.950 


68 










6.295 


88 


5.1 






6.426 


65 


7.632 


75 






5.2 


8.749 


85 










5.399 


73 


5.3 










7.026 


66 






5.7 


8.260 


50 










5.605 


78 


5.8 






5.729 


70 


8.646 


65 






5.9 


7.688 


67 














6.0 










7.177 


70 






6.1 






6.191 


65 






6.729 


65 


6.2 


9.666 


59 














6.3 










8.212 


70 






6.9 






5.895 


73 










7.1 














6.667 


65 


7.2 










10.018 


60 

























96 



97 



Table C-2. Dynamic Strength of Holed-Notched Cylinders, 102 mm (4 in) Diameter 



Strain Rate 


Group 5 


Group 6 


Group 7 


Group 8 


Peak Stress 


Time-to- 


Peak Stress 


Time-to- 


Peak Stress 


Time-to- 


Peak Stress 


Time-to- 


m/m/s 


MPa 


Failure, us 


MPa 


Failure, us 


MPa 


Failure, us 


MPa 


Failure, us 


1.7 






3.523 


105 










1.9 














3.875 


125 


2.0 


5.909 


85 














2.1 










5.467 


100 






2.2 


5.998 


120 










4.530 


110 


2.3 










4.730 


115 






2.6 






3.820 


100 










2.7 










5.626 


105 






3.0 






4.351 


100 


5.123 


95 






3.3 














3.916 


100 


3.7 


6.991 


80 














3.8 














4.261 


100 


3.9 










8.260 


95 






4.0 


7.550 


75 






7.598 


85 






4.1 














4.985 


90 


4.2 






5.095 


95 










4.3 










7.432 


85 






4.6 


7.688 


80 














4.7 














6.274 


90 


5.0 






6.012 


85 










5.1 














6.116 


85 


5.3 






5.723 


85 










5.7 


8.274 


65 






8.115 


75 






5.9 














6.329 


75 


6.4 






7.439 


80 










7.1 










8.012 


75 






7.3 














7.908 


70 






98 



Table C-3. Dynamic Strength of Holed-Notched Cylinders, 152 mm (6 in) Diameter 



Strain Rate 


Group 9 


Group 10 


Peak Stress 


Time-to- 


Peak Stress 


Time-to- 


m/m/s 


MPa 


Failure, us 


MPa 


Failure, ^s 


2.8 


7.274 


145 






3.0 






6.371 


125 


3.1 


6.881 


150 






3.3 






7.060 


145 


3.6 


7.984 


115 


7.260 


130 


3.7 


8.722 


115 






3.8 






7.495 


125 


3.9 


8.543 


100 


7.315 


105 


4.0 


9.053 


115 


7.750 


100 


4.1 


8.853 


105 






4.2 






7.384 


105 


4.3 


7.708 


100 


7.329 


110 


4.7 






7.674 


100 


4.8 


9.122 


85 






5.0 






8.150 


100 


5.2 


9.411 


85 







APPENDIX D 
SYNCHRONIZED STRESS HISTORY AND ULTRAHIGH- SPEED PHOTOGRAPHS 



This assemblage of experimental data relates the physical fracture event with the 
measured stresses for a variety of holed-notched cylinders. All data are reported with time 
referenced to the time when the incident wave reaches the interface of the load distributor 
and specimen surface on the side of the incident bar. Temporal referencing was performed 
by accounting for the time of travel of the elastic wave along appropriate lengths of the 
incident bar, specimen, and transmitter bar. The field-of-view of the ultra-high speed 
camera for each experiment was set to capture either of the two views described by Figure 
D-l. After the experiments, the photographs were digitally extracted and placed on the 
stress history graph using the appropriate time basis. 




Incident Bar 

J L 




View Region 1 



View Region 2 



Figure D-l . Regions Viewed by the Ultra-High Speed Camera 



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APPENDIX E 

INITIAL CALCULATIONS FOR THE TWO PARAMETER FRACTURE MODEL, AN 

EXAMPLE USING ONE STRAIN RATE, (RATE = 2/S) 



STRAI N RATE = 2 per sec 

Peak Load Calculations, Specific for Holed-Notched, t/R = 0.16, r/R=0.08 



GROUP PROPERTIES: 
E (psi) = 
Poisson's v = 



4.15E+06 2.86E+10 (Pa) 
0.2 



Peak Stress (psi), a = 

Max Load (lbs), P = 

Specimen Diam. (in), D = 

Specimen Length (in), b = 

Initial Notch Length (in), 2ao = 

"""iter, a/R = 



830.92 
5873 
3.00 

1.5 
0.750 



Listing of 
permissible crack 
lengths during the 
fracture process at 
peak stress 



pter, t/R = 

bm., r/R = 

i 

! (in), 2r = 



Stress intensity for 
the current crack 
length. 

Calculated using 
Equation 1 



26125 (N) 
76.2 (mm) 
38.1 (mm) 
19.1 (mm) 

0.250 
0.16 
0.08 
6.35 (mm) 



Description of 
Physical Properties 
and Specimen 
Geometry 



> 



Crack tip opening 
displacement for 
the current crack 
length. 

Calculated using 
Equation 1 



J 



a/R 



ao/a 



| K1 (MPei'm^.S)! 



( Coefficients required to 
L0 ^1 



cabulate CTOD ) 
L2 L3 



u 



CTOD (mm) 



0.250 
0.252 
0.253 
0.255 
0.257 
0.259 
0.260 
0.262 
0.264 
0.265 
0.267 
0.269 
0.270 
0.272 
0.274 
0.276 
0.277 



1.175 



I 



1.9725336 



»936 5.702638 -3.01902 



0.00126 



Aqsi. J 


1.182 


Ratio of initial to 


1.186 


current crack 




length, used for 


1.190 


crack tip opening 


1.193 


displacement 


1.197 


0.955 


1.201 


0.948 


1.204 


0.942 


1.208 



I 



1.179 1.9617072 -4522rt3 5.679196 -3.01244 0.00143 
\ ,| 

Coefficients for the crack tip 

opening displacement, Equation 20 I 



1.9299909 -4.44148 5.609422 -2.99274 

1.9196692 -4.41494 5.586347 -2.9862 

-4.38857 5.563, 

-4.36236 JW40472 

L 33632/ 5.51767 

-4.3\0/5 5494959 




0.00161 
0.00178 
0.00195 
0.00211 
0.00227 
0.00243 
0.00259 
0.00275 



0.936, 37j e5e f WO columns, K ; and CTOD, of each Group Type is 
0930 l curve-Jit to a 3rd or 4th order polynomial to provide the ability 
0.925! to evaluate all groups at common abscissa, K ; values. 

O.919"" " T 1 ~2~24 " " 1 1 84"14387" "-"4" 20864" ~5~4C)5"o"l3" " "-Z"934~18~|" " " 6 J30334~ " 

I I 

0.913 1.228 1.8321859 -4.18359 5.38275 -2.92773 | 0.00349 

0.907 1.232 1.8230462 -4.15871 5.360576 -2.92128 | 0.00363 

0.902 | 1.236 1.8140186 -4.134 5.33849 -2.91484 | 0.00377 

Values are calculated for the range of all possible notch lengths that could exist during the 
loading event; ranging from the smallest at initial notch length to the largest at the outer region 
of the inner 80% of the radius that is subject to mode I splitting tension. 

I I . 

■ ■ ■ ■ ■ ■ ■ 

■ I ■ ■ ■ ■ ■ ■ 

■ ■ ■ ■ ■ 



V_ 



y 



L.i_; 



Figure E-l . Explanation of Terms of the the Peak Load Method for the Two Parameter 

Fracture Model 



122 







123 












STRAIN RATE = 


2 


per sec 












Peak Load Calculations, Specific for Holed-Notched, t/R = 0.16, r/R=0.08 














GROUP PROPERTIES: 












E (psi) = 


4.15E+06 


2.86E+10 (Pa) 












Poisson's v = 


0.2 














Peak Stress (psi), o = 


830.92 










Max Load (lbs), P = 


5873 


26125 (N) 










Specimen Diam. (in), D = 


3.00 


76.2 (mm) 










Specimen Length (in), b = 


1.5 


38.1 ( 


mm) 










Initial Notch Length (in), 2ao = 


0.750 


19.1 { 


mm) 










Initial Notch/Diameter, a/R = 


0.250 


0.250 












Load Width/Diameter, t/R = 


0.16 


0.16 












Hole diam./Specimen Diam., r/R = 


0.08 


0.08 














Hole Diam. (in), 2r = 


0.250 


6.35 (mm) 






















( Coefficients required to calculate CTOD ) 




a/R 


ao/a 


K1 (MPa*m A 0.5) 


L0 


L1 


L2 


L3 


CTOD (mm) 




0.250 


1.000 


1.175 


1.9725336 


-4.54936 


5.702638 


-3.01902 


0.00126 


0.252 


0.993 


1.179 


1.9617072 


-4.52213 


5.679196 


-3.01244 


0.00143 


0.253 


0.987 


1.182 


1.9510087 


-4.49508 


5.655846 


-3.00586 


0.00161 


0.255 


0.980 


1.186 


1.940437 


-4.4682 


5.632588 


-2.9993 


0.00178 


0.257 


0.974 


1.190 


1.9299909 


-4.44148 


5.609422 


-2.99274 


0.00195 


0.259 


0.967 


1.193 


1.9196692 


-4.41494 


5.586347 


-2.9862 


0.00211 


0.260 


0.961 


1.197 


1.9094708 


-4.38857 


5.563364 


-2.97966 


0.00227 


0.262 


0.955 


1.201 


1 .8993946 


-4.36236 


5.540472 


-2.97314 


0.00243 


0.264 


0.948 


1.204 


1.8894394 


-4.33632 


5.51767 


-2.96662 


0.00259 


0.265 


0.942 


1.208 


1.879604 


-4.31045 


5.494959 


-2.96012 


0.00275 


0.267 


0.936 


1.212 


1.8698874 


-4.28475 


5.472338 


-2.95362 


0.00290 


0.269 


0.930 


1.216 


1.8602884 


-4.25921 


5.449807 


-2.94713 


0.00305 


0.270 


0.925 


1.220 


1.8508059 


-4.23384 


5.427365 


-2.94065 


0.00320 


0.272 


0.919 


1.224 


1.8414387 


-4.20864 


5.405013 


-2.93418 


0.00334 


0.274 


0.913 


1.228 


1.8321859 


-4.18359 


5.38275 


-2.92773 


0.00349 


0.276 


0.907 


1.232 


1.8230462 


-4.15871 


5.360576 


-2.92128 


0.00363 


0.277 


0.902 


1.236 


1.8140186 


-4.134 


5.33849 


-2.91484 


0.00377 


Figure E-2. 


Peak Load Calculations for Group 1, (Example using 76 mm Diam. Cylinder, 




a/R=0.25 and r/R=0.08 at Strain Rate 


= 2/s) 







124 



STRAIN RATE = 2 per sec 

Peak Load Calculations, Specific for Holed-Notched, t/R = 0.16, r/R=0.08 





GROUP PROPERTIES: 


















E (psi) = 


4.15E+06 


2.86E+10 (Pa) 










Poisson's v = 




0.2 














Peak Stress (psi), c = 


565.75 










Max Load (lbs), P = 




3999 


17788 (N) 










Specimen Diam. (in), D = 




3.00 


76.2 (mm) 








Specimen Length (in), b = 




1.5 


38.1 (mm) 








Initial Notch Length (in), 2ao = 


' 


1 .350 


34.3 (mm) 








Initial Notch/Diameter, a/R = 


0.450 


0.450 










Load Width/Diameter, t/R = 




0.16 


0.16 










Hole diam ./Specimen Diam ., r/R ■ 




0.08 


0.08 












Hole Diam. On), 2r = 


0.250 


6.35 (mm) 
























( Coefficients required to calculate CTOD ) 




a/R 


ao/a 


K1 (MPa'm' 


0.5) 


L0 


L1 


L2 


L3 


CTOD (mm) 


0.450 


1.000 


1.190 




1.3093223 


-2.34929 


3.505889 


-2.30983 


0.00238 


0.451 


0.997 


1.193 




1.3075313 


-2.34042 


3.494827 


-2.30565 


0.00250 


0.453 


0.994 


1.197 




1.3057569 


-2.3316 


3.483799 


-2.30148 


0.00261 


0.454 


0.991 


1.200 




1.3039988 


-2.32284 


3.472807 


-2.29731 


0.00272 


0.455 


0.989 


1.203 




1.3022567 


-2.31413 


3.46185 


-2.29315 


0.00283 


0.457 


0.986 


1.206 




1.3005304 


-2.30547 


3.450928 


-2.28899 


0.00294 


0.458 


0.983 


1.209 




1.2988198 


-2.29687 


3.440041 


-2.28484 


0.00304 


0.459 


0.980 


1.212 




1.2971245 


-2.28832 


3.429187 


-2.2807 


0.00315 


0.460 


0.977 


1.215 




1.2954444 


-2.27982 


3.418369 


-2.27656 


0.00326 


0.462 


0.975 


1.218 




1.2937792 


-2.27138 


3.407585 


-2.27242 


0.00337 


0.463 


0.972 


1.221 




1.2921287 


-2.26299 


3.396834 


-2.26829 


0.00347 


0.464 


0.969 


1.224 




1.2904928 


-2.25465 


3.386118 


-2.26417 


0.00358 


0.466 


0.966 


1.228 




1.288871 


-2.24636 


3.375436 


-2.26005 


0.00368 


0.467 


0.964 


1.231 




1.2872633 


-2.23812 


3.364788 


-2.25594 


0.00379 


0.468 


0.961 


1.234 




1.2856695 


-2.22993 


3.354174 


-2.25183 


0.00389 


0.470 


0.958 


1.237 




1.2840893 


-2.2218 


3.343593 


-2.24773 


0.00399 


0.471 


0.956 


1.240 




1.2825225 


-2.21371 


3.333045 


-2.24363 


0.00410 


0.472 


0.953 


1.243 




1.2809689 


-2.20568 


3.322531 


-2.23954 


0.00420 


0.473 


0.951 


1.246 




1.2794283 


-2.19769 


3.31205 


-2.23546 


0.00430 


0.475 


0.948 


1.250 




1.2779005 


-2.18975 


3.301602 


-2.23138 


0.00440 


0.476 


0.945 


1.253 




1.2763853 


-2.18187 


3.291187 


-2.22731 


0.00450 



Figure E-3. Peak Load Calculations for Group 2, (Example using 76 mm Diam. Cylinder, 

a/R=0.45 and r/R=0.08 at Strain Rate = 2/s) 









125 












STRAIN RATE = 


2 


per sec 












Peak Load Calculations, Specific for Holed-Notched, t/R = 


J.16, r/R=0.12 














GROUP PROPERTIES: 












E (psi) = 


4.47E+06 


3.08E+10 (Pa) 












Poisson's v = 


0.2 














Peak Stress (psi), o = 


723.45 










Max Load (lbs), P = 


5114 


22746 (N) 










Specimen Diam. (in), D = 


3.00 


76.2 (mm) 










Specimen Length (in), b = 


1.5 


38.1 (mm) 










Initial Notch Length (in), 2ao ■ 


0.750 


19.1 ( 


mm) 










initial Notch/Diameter, a/R = 


0.250 


0.250 












Load Width/Diameter, t/R = 


0.16 


0.16 












Hole diamiSpecimen Diam., r/R = 


0.13 


0.13 














Hole Diam. (in), 2r = 


0.375 


9.525 (mm) 






















( Coefficients required to calculate CTOD ) 




a/R 


ao/a 


K1 (MPaWO.S) 


L0 


L1 


L2 


L3 


CTOD (mm) 




0.250 


1.000 


1.225 


3.1444906 


-7.77503 


8.761129 


-4.19138 


-0.00064 


0.252 


0.993 


1.227 


3.1154595 


-7.69519 


8.675173 


-4.15679 


-0.00045 


0.253 


0.987 


1.229 


3.0868555 


-7.6165 


8.591186 


-4.12309 


-0.00026 


0.255 


0.980 


1.231 


3.058674 


-7.53896 


8.509128 


-4.09025 


-0.00008 


0.257 


0.974 


1.232 


3.0309106 


-7.46255 


8.428961 


-4.05825 


0.00011 


0.259 


0.967 


1.234 


3.0035609 


-7.38726 


8.350646 


-4.02708 


0.00029 


0.260 


0.961 


1.236 


2.9766205 


-7.31309 


8.274144 


-3.99671 


0.00048 


0.262 


0.955 


1.238 


2.9500849 


-7.24001 


8.199418 


-3.96714 


0.00066 


0.264 


0.948 


1.240 


2.9239499 


-7.16802 


8.12643 


-3.93834 


0.00085 


0.265 


0.942 


1.242 


2.898211 


-7.0971 


8.055143 


-3.9103 


0.00103 


0.267 


0.936 


1.244 


2.8728639 


-7.02725 


7.985522 


-3.883 


0.00121 


0.269 


0.930 


1.247 


2.8479044 


-6.95844 


7.91753 


-3.85641 


0.00139 


0.270 


0.925 


1.249 


2.8233282 


-6.89067 


7.851132 


-3.83054 


0.00157 


0.272 


0.919 


1.251 


2.799131 


-6.82393 


7.786293 


-3.80535 


0.00175 


0.274 


0.913 


1.253 


2.7753086 


-6.75821 


7.722979 


-3.78083 


0.00193 


0.276 


0.907 


1.256 


2.7518567 


-6.69348 


7.661156 


-3.75697 


0.00211 


0.277 


0.902 


1.258 


2.7287713 


-6.62975 


7.60079 


-3.73376 


0.00229 


0.279 


0.896 


1.260 


2.7060481 


-6.56699 


7.541848 


-3.71116 


0.00247 


0.281 


0.891 


1.263 


2.6836829 


-6.5052 


7.484298 


-3.68918 


0.00264 


0.282 


0.886 


1.265 


2.6616718 


-6.44437 


7.428107 


-3.6678 


0.00282 


0.284 


0.880 


1.268 


2.6400106 


-6.38449 


7.373245 


-3.64699 


0.00300 


Figure E-4. 


Peak Load Calculations for Gi 
a/R=0.25 and r/R=0 


oup 3, (E 
.12atStn 


sample 
dn Rate 


using 76 
= 2/s) 


mm Diam. Cylinder, 



126 



STRAIN RATE = 2 per sec 

Peak Load Calculations, Specific for Holed-Notched, t/R = 0.16, r/R=0.12 





GROUP PROPERTIES: 


















E(psi) = 


4.47E+06 


3.08E+10 (Pa) 










Poisson's v = 




0.2 














Peak Stress (psi), o = 


457.11 










Max Load (lbs), P = 




3231 


14372 (N) 










Specimen Diam. (in), D = 




3.00 


76.2 (mm) 








; 


Specimen Length (in), b = 




1.5 


38.1 (mm) 








Initial Notch Length (in), 2ao = 




1. 350 


34.3 (mm) 








Initial Notch/Diameter, a/R = 


I 


3.450 


0.450 










Load Width/Diameter, t/R = 




0.16 


0.16 










Hole diam./Specimen Diam., r/R = 




0.13 


0.13 












Hole Diam. (in), 2r = 


I 


3.375 


9.525 (mm) 
















( Coefficients required to calculate CTOD ) 




a/R 


ao/a 


K1 (MPa*m' 


US) 


L0 


L1 


L2 


L3 


CTOD (mm) 


0.450 


1.000 


1.048 




1.6470002 


-3.5281 


4.934769 


-2.80887 


0.00322 


0.451 


0.997 


1.050 




1.6445211 


-3.51966 


4.922868 


-2.80453 


0.00329 


0.452 


0.995 


1.052 




1.6420703 


-3.51128 


4.910943 


-2.80018 


0.00335 


0.454 


0.992 


1.055 




1.6396472 


-3.50298 


4.898996 


-2.7958 


0.00341 


0.455 


0.989 


1.057 




1.6372512 


-3.49473 


4.887024 


-2.7914 


0.00347 


0.456 


0.987 


1.059 




1.6348816 


-3.48655 


4.875029 


-2.78699 


0.00353 


0.457 


0.984 


1.062 




1.6325379 


-3.47844 


4.863009 


-2.78255 


0.00358 


0.458 


0.982 


1.064 




1.6302194 


-3.47038 


4.850966 


-2.7781 


0.00364 


0.460 


0.979 


1.066 




1.6279255 


-3.46238 


4.838897 


-2.77362 


0.00370 


0.461 


0.977 


1.069 




1.6256556 


-3.45443 


4.826805 


-2.76913 


0.00375 


0.462 


0.974 


1.071 




1.6234091 


-3.44654 


4.814688 


-2.76462 


0.00381 


0.463 


0.972 


1.073 




1.6211854 


-3.4387 


4.802546 


-2.76009 


0.00386 


0.464 


0.969 


1.076 




1.618984 


-3.43092 


4.790379 


-2.75553 


0.00391 


0.466 


0.966 


1.078 




1.6168042 


-3.42318 


4.778188 


-2.75096 


0.00397 


0.467 


0.964 


1.080 




1.6146455 


-3.41549 


4.765973 


-2.74638 


0.00402 


0.468 


0.962 


1.083 




1.6125073 


-3.40785 


4.753733 


-2.74177 


0.00407 


0.469 


0.959 


1.085 




1.610389 


-3.40025 


4.741469 


-2.73714 


0.00412 


0.470 


0.957 


1.087 




1.6082902 


-3.3927 


4.72918 


-2.7325 


0.00416 


0.472 


0.954 


1.090 




1.6062102 


-3.38519 


4.716868 


-2.72784 


0.00421 


0.473 


0.952 


1.092 




1.6041484 


-3.37772 


4.704533 


-2.72316 


0.00426 


0.474 


0.949 


1.095 




1.6021045 


-3.37029 


4.692174 


-2.71846 


0.00431 



Figure E-5. Peak Load Calculations for Group 4, (Example using 76 mm Diam. Cylinder, 

a/R=0.45 and r/R=0.12 at Strain Rate - 2/s) 



127 



STRAIN RATE = 2 per sec 

Peak Load Calculations, Specific for Holed-Notched, t/R = 0.16, r/R=0.08 





GROUP PROPERTIES: 


















E(psi) = 


4.15E+06 


2.86E+10 


Pa) 










Poisson;s v= 




0.2 














Peak Stress (psi), o = 




B60.9 










Max Load (lbs), P = 




8114 


36090 (N) 










Specimen Diam. (in), D = 




4.00 


101 .6 (mm) 










Specimen Length (in), b - 




1.5 


38.1 


mm) 








Initial Notch Length On), 2ao = 




1.000 


25.4 


mm) 








Initial Notch/Spec. Diam., a/R = 




0.250 


0.250 










Load Width/Specimen Diameter, t/R = 




0.16 


0.16 










Hole diamVSpecimen Diam., r/R = 




0.09 


0.09 












Hole Diam. (in),2r» 




0.375 


9.525 


mm) 
















( Coefficients required to calculate CTOD ) 




a/R 


ao/a 


K1 (MPa*m*0.5) 


L0 


L1 


L2 


L3 


CTOD (mm) 


0.250 


1.000 


1.406 




1.9725336 


-4.54936 


5.702638 


-3.01902 


0.00173 


0.252 


0.993 


1.410 




1.9617072 


-4.52213 


5.679196 


-3.01244 


0.00198 


0.253 


0.987 


1.414 




1.9510087 


-4.49508 


5.655846 


-3.00586 


0.00222 


0.255 


0.980 


1.419 




1.940437 


-4.4682 


5.632588 


-2.9993 


0.00246 


0.257 


0.974 


1.423 




1 .9299909 


-4.44148 


5.609422 


-2.99274 


0.00269 


0.259 


0.967 


1.428 




1.9196692 


-4.41494 


5.586347 


-2.9862 


0.00292 


0.260 


0.961 


1.432 




1.9094708 


-4.38857 


5.563364 


-2.97966 


0.00314 


0.262 


0.955 


1.436 




1.8993946 


-4.36236 


5.540472 


-2.97314 


0.00336 


0.264 


0.948 


1.441 




1.8894394 


-4.33632 


5.51767 


-2.96662 


0.00358 


0.265 


0.942 


1.446 




1.879604 


-4.31045 


5.494959 


-2.96012 


0.00379 


0.267 


0.936 


1.450 




1.8698874 


-4.28475 


5.472338 


-2.95362 


0.00400 


0.269 


0.930 


1.455 




1.8602884 


-4.25921 


5.449807 


-2.94713 


0.00421 


0.270 


0.925 


1.459 




1.8508059 


-4.23384 


5.427365 


-2.94065 


0.00442 


0.272 


0.919 


1.464 




1.8414387 


-4.20864 


5.405013 


-2.93418 


0.00462 



Figure E-6. Peak Load Calculations for Group 5, (Example using 102 mm Diam. 
Cylinder, a/R=0.25 and r/R=0.08 at Strain Rate = 2/s) 



128 



STRAIN RATE = 2 per sec 

Peak Load Calculations, Specific for Holed-Notched, t/R = 0.16, r/R=0.08 



GROUP PROPERTIES: 






E(psi) = 


4.15E+06 


2.86E+10(Pa) 


Poisson.s v= 


0.2 




Peak Stress (psi), o « 


501.48 




Max Load (lbs), P = 


4726 


21023 (N) 


Specimen Diam. (in), D = 


4.00 


101 .6 (mm) 


Specimen Length (in), b = 


1.5 


38.1 (mm) 


Initial Notch Length (in), 2ao - 


1.800 


45.7 (mm) 


Initial Notch/Spec. Diam., a/R ■ 


0.450 


0.450 


Load Width/Specimen Diameter, t/R = 


0.16 


0.16 


Hole diam./Specimen Diam, r/R = 


0.09 


0.09 


Hole Diam. (in), 2r = 


0.375 


9.525 (mm) 



a/R 



ao/a 



K1 (MPa'm A 0.5) 



{ Coefficients required to calculate CTOD) 
L0 L1 L2 L3 



CTOD (mm) 



0.450 
0.451 
0.453 
0.454 
0.455 
0.457 
0.458 
0.459 
0.460 
0.462 
0.463 
0.464 
0.466 
0.467 



1.000 
0.997 
0.994 
0.991 
0.989 
0.986 
0.983 
0.980 
0.977 
0.975 
0.972 
0.969 
0.966 
0.964 



1.218 
1.222 
1.225 
1.228 
1.231 
1.234 
1.237 
1.240 
1.244 
1.247 
1.250 
1.253 
1.256 
1.260 



1.3093223 
1.3075313 
1.3057569 
1.3039988 
1.3022567 
1.3005304 
1.2988198 
1.2971245 
1.2954444 
1.2937792 
1.2921287 
1.2904928 
1.288871 
1.2872633 



-2.34929 
-2.34042 
-2.3316 
-2.32284 
-2.31413 
-2.30547 
-2.29687 
-2.28832 
-2.27982 
-2.27138 
-2.26299 
-2.25465 
-2.24636 
-2.23812 



3.505889 
3.494827 
3.483799 
3.472807 
3.46185 
3.450928 
3.440041 
3.429187 
3.418369 
3.407585 
3.396834 
3.386118 
3.375436 
3.364788 



-2.30983 
-2.30565 
-2.30148 
-2.29731 
-2.29315 
-2.28899 
-2.28484 
-2.2807 
-2.27656 
-2.27242 
-2.26829 
-2.26417 
-2.26005 
-2.25594 



0.00282 
0.00295 
0.00308 
0.00321 
0.00334 
0.00347 
0.00360 
0.00373 
0.00385 
0.00398 
0.00410 
0.00423 
0.00435 
0.00448 



Figure E-7. Peak Load Calculations for Group 6, (Example using 102 mm Diam. 
Cylinder, a/R=0.45 and r/R-0.08 at Strain Rate = 2/s) 





129 










STRAIN RATE = 


2 


per sec 












Peak Load Calculations, Specific for Holed-Notched, t/R = 0.16, 


/R=0.12 












GROUP PROPERTIES: 










E(psi) = 


4.15E+06 


2.86E+10(Pa) 










Poisson;s v= 


0.2 












Peak Stress (psi), a = 


779.68 








Max Load (lbs), P = 


7348 


32685 (N) 










Specimen Diam. On), D ■ 


4.00 


101.6 (mm) 










Specimen Length (in), b - 


1.5 


38.1 (mm) 










Initial Notch Length (in), 2ao ■ 


1.000 


25.4 (mm) 










Initial Notch/Spec. Diam., a/R = 


0.250 


0.250 












Load Width/Specimen Diameter, t/R = 


0.16 


0.16 












Hole diamVSpecimen Diam., r/R = 


0.13 


0.13 












Hole Diam. (in), 2r = 


0.500 


12.7 (mm) 












( Coefficients required to calculate CTOD ) 






a/R ao/a 


K1 (MPaWO.5) 


L0 


L1 


L2 


L3 


CTOD (mm) 




0.250 1.000 


1.525 


3.1444906 


-7.77503 


8.761129 


-4.19138 


-0.00099 


0.252 0.993 


1.527 


3.1154595 


-7.69519 


8.675173 


-4.15679 


-0.00070 


0.253 0.987 


1.529 


3.0868555 


-7.6165 


8.591186 


-4.12309 


-0.00041 


0.255 0.980 


1.531 


3.058674 


-7.53896 


8.509128 


-4.09025 


-0.00012 


0.257 0.974 


1.534 


3.0309106 


-7.46255 


8.428961 


-4.05825 


0.00017 


0.259 0.967 


1.536 


3.0035609 


-7.38726 


8.350646 


-4.02708 


0.00046 


0.260 0.961 


1.538 


2.9766205 


-7.31309 


8.274144 


-3.99671 


0.00074 


0.262 0.955 


1.541 


2.9500849 


-7.24001 


8.199418 


-3.96714 


0.00103 


0.264 0.948 


1.543 


2.9239499 


-7.16802 


8.12643 


-3.93834 


0.00131 


0.265 0.942 


1.546 


2.89821 1 


-7.0971 


8.055143 


-3.9103 


0.00159 


0.267 0.936 


1.549 


2.8728639 


-7.02725 


7.985522 


-3.883 


0.00187 


0.269 0.930 


1.551 


2.8479044 


-6.95844 


7.91753 


-3.85641 


0.00215 


0.270 0.925 


1.554 


2.8233282 


-6.89067 


7.851132 


-3.83054 


0.00243 


0.272 0.919 


1.557 


2.799131 


-6.82393 


7.786293 


-3.80535 


0.00271 


Figure E-8. Peak Load Calculations for Group 7, (Example using 102 mm Diam. 
Cylinder, a/R=0.25 and r/R=0.12 at Strain Rate - 2/s) 



130 



STRAIN RATE = 2 per sec 

Peak Load Calculations, Specific for Holed-Notched, t/R = 0.16, r/R=0.12 



GROUP PROPERTIES: 






E(psi) = 


4.15E+06 


2.86E+10(Pa) 


Poisson;s v= 


0.2 




Peak Stress (psi), o = 


534.29 




Max Load (lbs), P = 


5036 


22398 (N) 


Specimen Diam. (in), D = 


4.00 


101 .6 (mm) 


Specimen Length (in), b - 


1.5 


38.1 (mm) 


Initial Notch Length (in), 2ao = 


1.800 


45.7 (mm) 


Initial Notch/Spec. Diam., a/R = 


0.450 


0.450 


Load Width/Specimen Diameter, t/R = 


0.16 


0.16 


Hole diamVSpecimen Diam., r/R = 


0.13 


0.13 


Hole Diam. (in),2r = 


0.500 


12.7 (mm) 



a/R 



ao/a 



K1 (MPa'rr^O.S) 



( Coefficients required to calculate CTOD) 
L0 L1 L2 L3 



CTOD (mm) 



0.450 
0.451 
0.452 
0.454 
0.455 
0.456 
0.457 
0.458 
0.460 
0.461 
0.462 
0.463 
0.464 
0.466 



1.000 
0.997 
0.995 
0.992 
0.989 
0.987 
0.984 
0.982 
0.979 
0.977 
0.974 
0.972 
0.969 
0.966 



1.414 
1.417 
1.421 
1.424 
1.427 
1.430 
1.433 
1.436 
1.439 
1.442 
1.445 
1.449 
1.452 
1.455 



1.6470002 
1.6445211 
1.6420703 
1.6396472 
1.6372512 
1.6348816 
1.6325379 
1.6302194 
1.6279255 
1.6256556 
1.6234091 
1.6211854 
1.618984 
1.6168042 



-3.5281 
-3.51966 
-3.51128 
-3.50298 
-3.49473 
-3.48655 
-3.47844 
-3.47038 
-3.46238 
-3.45443 
-3.44654 

-3.4387 
-3.43092 
-3.42318 



4.934769 
4.922868 
4.910943 
4.898996 
4.887024 
4.875029 
4.863009 
4.850966 
4.838897 
4.826805 
4.814688 
4.802546 
4.790379 
4.778188 



-2.80887 

-2.80453 

-2.80018 

-2.7958 

-2.7914 

-2.78699 

-2.78255 

-2.7781 

-2.77362 

-2.76913 

-2.76462 

-2.76009 

-2.75553 

-2.75096 



0.00541 
0.00552 
0.00562 
0.00572 
0.00582 
0.00592 
0.00602 
0.00611 
0.00621 
0.00630 
0.00639 
0.00648 
0.00657 
0.00666 



Figure E-9. Peak Load Calculations for Group 8, (Example using 102 mm Diam. 
Cylinder, a/R-0.45 and r/R=0.12 at Strain Rate = 2/s 



APPENDIX F 
SPREADSHEET ANALYSIS FOR THE TWO PARAMETER FRACTURE MODEL 



76 mm Diam. Holed- Notched Data 



Group 1 



Group 2 



Group 3 



Group 4 



-0.0682419 
00262427 



0.00484209 




102 mm Diam. Holed-Notched Data 



Group 5 



Group 6 



Group 7 



Group 8 



Polynomial coefficients to curve fitting of individual group CTOD-K, 
data pairs. Used to get all groups at equivalent abscissa values. 

CTOD = a + bK,+ c{K, f + d(K { f + e(K l ) 4 



K1 
(MPa-m"0.5) 



CTOO Grp 1 CTOD Grp 2 CTOD Gp 3 CTOD Grp 4 CTOD Grp 5 CTOD Grp 6 CTOD Gp 7 CTOD Grp 8 
(mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) 



CT0D-(a\Q) 
(mm) 



Minimized 
Error. (mm) A 2 



1 .5200 
1 .5207 
1.5213 
1.5220 
1.5226 
1 .5233 
1.5239 
1 .5246 
1 .5252 
1 .5259 
1.5265 
1 .5272 
1 .5278 
1 .5285 
1.5291 
1 .5298 
1.5304 
1 .531 1 
1.5317 
1 .5324 



0.007889 
0.007906 
0.007922 
0.007938 
0.007955 
0.007971 
0.007987 



0.009244 
0.009260 
0.009277 
0.009293 
0.009310 
0.009326 
0.009342 



0.011169 
0.011190 
0.011210 
0.011230 
0.011251 
0.011271 
0.011291 



oijj>at cnoficvu ruoufifci Qi«ru£3...Ar^aiUi -ruu\o 504021 

i i 

0.01J Error minimization for establishing the mutual (273 
o.oi| solution to the eight types of cylinders all ' , i / mi 

0013 subjected to the same strain rate [03574 



0.012928 
0.012951 
0.012974 



0.006611 
0.006631 
0.006652 



0.012720 
0.012737 
0.012754 



-0. 000213 
-0.000153 
-0.000094 



°i Now that each group's CTOD-^ data have been curve 
I fit, we evaluate each group and the same abscissa value. 

0.00 or 

J Calculate the average for all groups and then minimize 
orj the error via Equation (6) 

0.008085 0.009441 0.011411 0.013115 0.006773 0.012857 0.000260 

0.008101 0.009457 0.011431 0.013139 0.006793 0.012874 0.000318 

0.008117 0.009473 0.011451 0.013163 0.005813 0.012891 0.000377 

0.008134 0.009490 0.011470 0.013187 0.006834 0.012908 0.000435 

0.008150 0.009506 0.011490 0.013210 0.006854 0.012925 0.000493 

0.008166 0.009522 0.011510 0.013234 0.006874 0.012942 0.000551 

0.008182 0.009539 0.011529 0.013258 0.006894 0.012959 0.000609 

0.008198 0.009555 0.011549 0.013282 0.006914 0.012976 0.000667 



0.008217 
0.008230 
0.008243 

66 

h 

)S2 
jM 

F 

0008320 
0.008332 
0.008345 
0.008358 
0.008370 
0.008383 
0.008395 
0.008408 



0.00859728 
0.008620535 
0.008643768 
0.008666977 
0.008690163 
0.008713326 
0008736466 
0.008759583 
0.008782677 
0.008805749 
0.008828798 
008851824 
0.008874827 
0.008897809 
0.008920768 
0.008943705 



i Values are calculated for the minimum K, range of all 8 groups to 
! extrapolation of the curve fits. 

! In other words, the range of possible values is defined and limited 
■ shared by all groups. 



iS^gJ 


0.058"2"95"" 


"o'aSSSi" 


"0.5ff6§5" 


"0"0'i5427"" 


"0" 007555" 


1.5369 


0.008311 


0.009668 


0.011684 


0.013452 


0.007055 


1 .5376 


0.008328 


0.009685 


0.011703 


0.013476 


0.007075 



1.8353 
1.8359 
1.8366 
1 .8372 
1 .8379 



0.015058 


0.016471 


0.017316 


0.028024 


0.015411 


0.015072 


0.016487 


0.017325 


0.028056 


0.015428 


0.015085 


0016502 


0.017334 


0.028087 


0.015444 


0.015099 


0.016517 


0017343 


0.028119 


0.015461 


0.015112 


0016532 


0.017352 


0.028151 


0.015477 



" 5 .WSbTB ~b~dol 5i"2" 

0.013095 0.001069 

0.013112 0.001126 



0.021148 0.018034 

0.021169 0.018056 

0.021189 0.018078 

0.021209 0.018100 

0.021230 0.018121 



prevent from 
to that which is 

' "0 5084S2l"0.509086"863 



0.008494 
0.008507 



0.009103647 
0.009126409 



0.014797 


0.018282553 


0014818 


0.018301172 


0.014839 


0.018319788 


0.014860 


0.018338401 


0.014881 


0.018357011 



S.9894E-05 

3.9623E-05 

3.9355E-05 

3.9089E-05 

3.8824E-05 

3.8561E-05 

3.83O0E-O5 

3.8040E-05 

3.7782E-05 

3.7527E-05 

3.7272E-05 

3.7020E-05 

3.6769E-05 

3.6520E-05 

3.6273E-05 

3.6027E-05 

3.5783E-05 

3.5541E-05 

3.5300E-05 

3.5061E-O5 

£-05 
i 

£-05 

r 

p-05 

f-05 

E-05 
i 

'3.3g63E-05 
3.3435E-05 
3.3209E-05 



1.3242E-05 
1.3263E-05 
1.3285E-05 
1.3306E-05 
1 3328E-05 



Figure F-l . Explanation of the Algorithm used to find a Mutual Solution to CTOD versus 

Kj c data through Error Minimization 



132 



133 



Coefficient 



76 mm Diam. Holed-Notched Data 



Group 1 



Group 2 



Group 3 



Group 4 



-0.0682419 -0.168236 

0.0862427 0.273838 

-0.031147 -0.14658 

0.00484209 0.0284481 





-0.651375 -0.863368 

1.26902 2.39375 

-0.907347 -2.45897 

0.287108 1.10991 

-0.0334023 -0.183382 



102 mm Diam. Holed-Notched Data 



Group 5 



Group 6 



Group 7 



Group 8 



-0.0843464 -0.182054 

0.0995844 0.316201 

-0.0335999 -0.180605 

0.00487986 0.0374023 





-0 953469 -1.34541 

1.57833 2.9772 

-095886 -2.44092 

0.257798 0.878952 

-0.0254838 -0.115958 



K1 


CTOD Grp 1 


CTOD Grp 2 


CTOD Gp 3 


CTOD Grp 4 


CTOD Grp 5 


CTOD Grp 6 


CTODGp7 


CTOD Grp 8 


CTOD-(avg) 


Minimized 


(MPa-m«0.5) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


Error, (mm) A 2 


1.5200 


0.007889 


0.009244 


0.011169 


0.012835 


0.006530 


0.012651 


-0.000452 


0.008165 


0.008504021 


3.9894E-05 


1.5207 


0.C07906 


0.009260 


0.011190 


0.012858 


0.006550 


0.012669 


-0.000392 


0.008178 


0.008527371 


3.9623E-05 


1.5213 


0.007922 


0.009277 


0.011210 


0.012881 


0.006570 


0.012686 


-O.000332 


0.008191 


0.008550697 


3.9355E-05 


1.5220 


0.007938 


0009293 


0.011230 


0.012904 


0.006591 


0.012703 


-0.000272 


0.008204 


0.008574 


3.9089E-05 


1.5226 


0.007955 


0.009310 


0.011251 


0.012928 


0.006611 


0.012720 


-0.000213 


0.008217 


000859728 


3.8824E-05 


1.5233 


0.007971 


0.009326 


0.011271 


0.012951 


0.006631 


0.012737 


-0.000153 


0.008230 


0.008620535 


3.856 1E-05 


1.5239 


0.007987 


0.009342 


0.011291 


0.012974 


0.006652 


0.012754 


-0.000094 


0.008243 


0.00864?765 


3.8300E-05 


1.5246 


0.008004 


0.009359 


0.011311 


0.012998 


0.006672 


0.012771 


-0.000035 


0.008256 


0.008666977 


3.3040E-05 


1.5252 


0.008020 


0.009375 


0.011331 


0.013021 


0.006692 


0.012788 


0.000024 


0.008269 


0.008690163 


3.7782E-05 


1.5259 


0.008036 


009392 


0.011351 


0.013045 


0.006712 


0.012806 


0.000083 


0.008282 


0.008713326 


3.7527E-05 


1.5265 


0.008052 


0.009408 


0.011371 


0.013068 


0.006733 


0.012823 


0.000142 


0.008294 


0.008736466 


3.7272E-05 


1.5272 


0.008069 


0.009424 


0.011391 


0.013092 


0.006753 


0.012840 


0.000201 


0.008307 


0008759583 


3.7020E-05 


1.5278 


0.008085 


0.009441 


0.011411 


0.013115 


0.006773 


0.012857 


0.000260 


0.008320 


0.008782677 


3.6769E-05 


1.5285 


0.008101 


0.009457 


0.011431 


0.013139 


0.006793 


0.012874 


0.000318 


0.008332 


0.008805749 


3.6520E-05 


1.5291 


0.008117 


0.009473 


0.011451 


0.013163 


0.006813 


0.012891 


0.000377 


0.008345 


0.008828798 


3.6273E-05 


1.5298 


0.008134 


0.009490 


0.011470 


0.013187 


0.006834 


0.012908 


0.000435 


0.008358 


0.008851824 


3.6027E-05 


1.5304 


0.008150 


0009506 


0.011490 


0.013210 


0.006854 


0.012925 


000493 


0.008370 


0.008874827 


3.5783E-05 


1.5311 


0.008166 


0.009522 


0.011510 


0.013234 


0.006874 


0.012942 


0.000551 


0.008383 


0.008897809 


3.5541E-05 


1.5317 


0.008182 


0.009539 


0.011529 


0.013258 


0.006894 


0.012959 


0.000609 


0.008395 


0.0C8920766 


3.6300E-05 


1.5324 


0.008198 


0.009555 


0.011549 


0.013282 


0.006914 


0.012976 


0.000667 


0.008408 


0.008943705 


3.5061E-05 


1.5330 


0.008214 


0.009571 


0.011568 


0.013306 


0.006934 


0.012993 


0.000725 


0.008420 


0.008966619 


3.4824E-05 


1.5337 


0.008231 


0.009587 


0.011588 


0.013330 


0.006954 


0.013010 


0.000783 


0.008433 


0.008989512 


3.4589E-05 


1.5343 


0.008247 


0.009604 


0.011607 


0.013355 


0.006975 


0.013027 


0.000840 


0.008445 


0.009012382 


3.4355E-05 


1.5350 


0.008263 


0.009620 


0.011627 


0.013379 


0.006995 


0.013044 


0.000898 


0.008457 


0.009035231 


3.4122E-05 


1.5356 


0.008279 


0.009636 


0011646 


0013403 


0.007015 


0.013061 


0.000955 


0.008470 


0.009058058 


3 3892E-05 


1.5363 


0008295 


0.009652 


0.011665 


0.013427 


0.007035 


0.013078 


0.001012 


0.008482 


0.009080863 


3.3663E-05 


1.5369 


0.0O331 1 


0.009668 


0.011684 


0.013452 


0.007055 


0.013095 


0.001069 


0.008494 


0.009103647 


3.3435E-05 


1.5376 


0.008328 


0.009685 


0.011703 


0.013476 


0.007075 


0.013112 


0.001126 


0.008507 


0.009126409 


3.3209E-05 



1.3340 


0.015031 


0.016441 


0.017298 


0.027960 


0.015378 


0.021108 


0.017990 


0.014756 


0.018245305 


1.3199E-05 


1.8346 


0.015045 


0.016456 


0.017307 


0.027992 


0.015394 


0.021128 


0.018012 


0.014776 


0.018263931 


1.3220E-05 


1.8353 


0015058 


0.016471 


0.017316 


0.028024 


0.015411 


0.021148 


0.018034 


0.014797 


0.018282553 


1.3242E-05 


1.8359 


0.015072 


0.016487 


0.017325 


0.028056 


0.015428 


0.021 169 


0.018056 


0.014818 


0.018301172 


1.3263E-05 


1.8366 


0.015085 


0.016502 


0.017334 


0.028087 


0.015444 


0.021 189 


0.018078 


0.014839 


0.018319788 


1.3285E-05 


1.8372 


0.015099 


0.016517 


0.017343 


0.028119 


0.015461 


0.021209 


0.018100 


0.014860 


0.018338401 


1.3306E-05 


1.8379 


0.015112 


0.016532 


0.017352 


0028151 


0.015477 


0.021230 


0.018121 


0.014881 


0.018357011 


1.3328E-05 



Figure F-2. Results of Curve Fit CTOD -vs- K ]c Data and Solution by Error 
Minimization, Strain Rate = 2 /s 



134 



0.035 

0.030^ 

_ 0.025- 

E 

E. 0.020 

O 0.015 
P 

° 0.010 
0.005- 



0.000 



Group # and Spec/men Type 

Group 1 , a/R=0.08, r/R=0.25 

— Group 2, a/R=0.08, r/R=0.45 

A Group 3, a/R=0.12, r/R=0.25 

— | Group 4. a/R=0.12, r/R=0.45 




1.00 1.20 1.40 1.60 1.80 2.00 2.20 
K1c(MPa-m A 0.6) 



2.40 2.60 



2.80 



E 

E, 

o 

a 
o 

i- 
o 



0.045 
0.040 
0.035 
0.030 
0.025 
0.020 
0.015 
0.010 
0.005 
0.000 



Group # and Specimen Type 

Group 5, a/R=0.08, r/R=0.25 

- F3 — Group 6, a/R=0.08, r/R=0.45 
-^ — Group 7, a/R=0.12, r/R=0.25 
-))( — Group 8, a/R=0.12, r/R=0.45 




Rate ■ 2/s 



I I I ♦ I I I I I I I I I I I I I 

1.20 1.40 1.60 1.80 2.00 2.20 2.40 
K1c(MPa-m A 0.5) 



i i i i | i i i i | i i i i i i i i i i i i i i i 
2.60 2.80 3.00 



N 3.00E-05 



.5. 2.50E-05 

t? 
O 



UJ 

ai 



2.00E-05 



1.50E-05 - 



g 1.00E-05 I i i i i I i i i i i i i i i i 



i i i i i 



1.50 1.55 1.60 1.65 1.70 1.75 

K1c (MPa-m A 0.5) 



-i— i — i i i i i i 

1.80 1.85 



Figure F-3. Two Parameter Fracture Model, Strain Rate = 2 /s 
Top: Curve Fit CTOD -vs- K lc Data Groups 1-4 
Middle: Curve Fit CTOD -vs- K Ic Data Groups 5-8 
Bottom: Solution by Error Minimization 



135 



Coefficient 



76 mm Diam. Ho led- Notched Data 



Group 1 



Group 2 



Group 3 



Group 4 



-0.0751227 -0.187025 

0.0862428 0.273837 

-0.0282942 -0.131853 

0.0039957 0.0230191 



-0.750576 -1.0496 

1.26902 2.39375 

-0.787426 -2.02268 

0.216231 0.750657 

-0.0218315 -0.102066 



102 mm Diam Holed-Notched Data 



Group 5 
-0.0938129 

0.0995846 

-0.0302096 

0.00394474 





Group 6 

-0.22565 

0.316201 

-0.145712 

0.0243459 





Group 7 



-1.07776 

1.57833 

-0.848278 

0.201765 

-0.0176446 -0.0674703 



Group 8 
-1.61157 
2.9772 
-2.03778 
0.612595 



K1 


CTOD Grp 1 


CTOD Grp 2 


CTODGp3 


CTOD Grp 4 


CTOD Grp 5 


CTOD Grp 6 


CTODGp7 


CTOD Grp 8 


CTOD-(avg) 


Minimized 


(MPa-m*05) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


Error, (mm) A 2 


1.7200 


0.009841 


0.011032 


0.011825 


0010138 


0.008173 


0.011024 


-0.000335 


0.007295 


0.008624155 


3.9809E-05 


1.7210 


0.009866 


0.011057 


0.011860 


0.010158 


0.008204 


0.011055 


-0.000243 


0.007325 


0.00866023 


3.9395E-05 


1.7220 


0.009890 


0.011081 


0.011895 


0.010179 


0.008234 


0.011086 


-0.000152 


0007356 


0.008696239 


3.8986E-05 


1.7230 


0.009914 


0.011106 


0.011930 


0.010200 


0.008265 


0.011117 


-0.000061 


0.007387 


0.008732183 


3.8579E-05 


1.7240 


0.009939 


0.011130 


0.011964 


0.010220 


0.008296 


0.011148 


0.000030 


0007417 


0.008768061 


3.8177E-05 


1.7250 


0.009963 


0.011155 


0.011999 


0.010241 


0.008326 


0.011179 


0.000121 


0.007447 


0.008803874 


3.7779E-05 


1.7260 


0.009987 


0.011179 


0.012033 


0.010262 


0.008357 


0.011210 


0.000211 


0.007477 


0.008839621 


3.7384E-05 


1.7270 


0.010012 


0.011204 


0.012068 


0.010283 


0.008387 


0.011240 


0.000302 


0.007507 


0.008875305 


3.6993E-05 


1.7280 


0.010036 


0.011228 


0.012102 


0.010304 


0.008418 


0.011271 


0.000392 


0.007537 


0.008910923 


3.6606E-05 


1.7290 


0.010060 


0.011252 


0.012136 


0.010326 


0.008448 


0.011302 


0.000482 


0.007566 


0.008946478 


3.6222E-05 


1.7300 


0.010084 


0.011277 


0.012170 


0.010347 


0.008479 


0.011332 


0.000571 


0.007596 


0.008981969 


3.5842E-05 


1.7310 


0.010108 


0.011301 


0.012203 


0.010368 


0.008509 


0.011363 


0.000660 


0.007625 


0.009017397 


3.5466E-05 


1.7320 


0.010133 


0.011325 


0.012237 


0.010390 


0008540 


0.011394 


0.000750 


0.007654 


0.009052761 


3.5094E-05 


1.7330 


0.010157 


0.011349 


0.012271 


0.010411 


0.008570 


0.011424 


0.000838 


0.007684 


0.009088063 


3.4725 E-05 


1.7340 


0.010181 


0.011374 


0.012304 


0.010433 


0.008601 


0.011455 


0.000927 


0.007712 


0.009123302 


3.4359E-05 


1.7350 


0.010205 


0.011398 


0.012337 


0.010455 


0.008631 


0.011485 


0.001015 


0.007741 


0009158478 


3.3997E-05 


1.7360 


0.010229 


0.011422 


0.012371 


0.010476 


0.008661 


0.011516 


0.001104 


0.007770 


0.009193593 


3.3639E-05 


1.7370 


0.010253 


0.011446 


0.012404 


0.010498 


0.008692 


0.011546 


0.001191 


0.007798 


0.009228646 


3.3284E-05 


1.7380 


0.010278 


0.011470 


0.012437 


0.010520 


0.008722 


0.011576 


0.001279 


0.007827 


0009263638 


3.2933E-05 


1.7390 


0.010302 


0.011494 


0.012470 


0.010542 


0.008752 


0.011607 


0.001367 


0.007855 


0.009298568 


3.2585E-05 


1.7400 


0.010326 


0.011518 


0.012503 


0.010564 


0.008783 


0.011637 


0.001454 


0.007883 


0.009333437 


3.2240E-05 


1.7410 


0.010350 


0.011542 


0.012535 


0010586 


0.008813 


0.011667 


0.001541 


0.007911 


0.009368246 


3.1899E-05 


1.7420 


0.010374 


0.011566 


0.012568 


0.010609 


0.008843 


0.011697 


0.001628 


0.007939 


0.009402995 


3.1561 E-05 


1.7430 


0.010398 


0.011591 


0.012600 


0.010631 


0.008873 


0.011727 


0.001714 


0.007967 


0.009437684 


3.1227E-05 


1.7440 


0.010422 


0.011614 


0.012633 


0.010653 


0.008904 


0.011758 


0.001800 


0.007994 


0.009472313 


3.0896E-05 


1.7450 


0.010446 


0.011638 


0.012665 


0.010676 


0008934 


0.011788 


0.001886 


0.008022 


0.009506882 


3.0568E-05 


1.7460 


0.010470 


0.011662 


0.012697 


0.010699 


0.008964 


0.011818 


0001972 


0008049 


0.009541393 


3.0244E-05 


1.7470 


0.010494 


0.011686 


0.012729 


0.010721 


0.008994 


0.011848 


0.002058 


0.008077 


0.009575844 


2.9922E-05 



2.2030 


0.020273 


0.022439 


0.021180 


0.029056 


0.021134 


0.024067 


0.024031 


0.017891 


0.022508922 


1.2412E-05 


2.2040 


0.020293 


0.022467 


0.021 195 


0.029102 


0.021158 


0.024096 


0.024054 


0.017923 


0.022536044 


1.2416E-05 


2.2050 


0.020313 


0.022495 


0.021209 


029148 


0.021182 


0.024124 


0.024078 


0.017956 


0.022563171 


1.2421E-05 


2.2060 


0.020332 


0.022524 


0.021223 


0.029194 


0.021206 


0.024153 


0.024102 


0.017989 


0.022590302 


1.2426E-05 


2.2070 


0.020352 


0.022552 


0.021237 


0.029240 


0.021230 


0.024182 


0.024126 


0.018021 


0022617439 


1.2430E-05 


2.2080 


0.020372 


0.022580 


0.021252 


0.029286 


0.021254 


0.024211 


0.024149 


0.018054 


0.02264458 


1 .2434E-05 


2.2090 


0.020391 


0022608 


0.021266 


0.029331 


0.021277 


0.024240 


0.024173 


0.018087 


0.022671726 


1.2438E-05 







Figure F-4. Results of Curve Fit CTOD -vs- K Ie Data and Solution by Error 
Minimization, Strain Rate « 3 /s 



136 



E 
E, 

o 
O 

o 

r- 
o 



0.045 

0.040 -= 

0.035 

0.030 

0.025^ 

0.020 

0.015 -g 

0.010 

0.005 



Group # and Specimen Type 
Group 1 , a/R=0.08, r/R=0.25 

(9 — Group 2, a/R=0.08, r/R=0.45 
-A — Group 3, a/R=0.12, r/R=0.25 

-|- Group 4, a/R=0.12, r/R=0.45 



0.00 

1.20 1.40 




i i i i i i i i i i i i 

1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 
K1c(MPa-m A 0.5) 



E 
E, 

o 
□ 
O 

I- 

o 



0.055 
0.050 
0.045 
0.040 
0.035 
0.030 
0.025 
0.020 
0.015 
0.010 
0.005 
0.000 



Group p a.nd Specimen Type 

Group 5, a/R=0.08, r/R=0.25 

- R — Group 6, a/R=0.08, r/R=0.45 
-^ — Group 7,a/R=0.12, r/R=0.25 
-# — Group8,a/R=0.12, r/R=0.45 




1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 
K1c (MPa-m A 0.6) 



cm 1.70E-05 

< 

£ 1.50E-05 



2 1.30E-05 ;; 

UJ 

T, 1.10E-05 

0) 

1 9.00E-06 :: 



Jg 7.00E-06 I i i i i i i i m ii i i i i i i i ii i i i i i i i i i i i i i i i i i i i i i i i i | i i i i 

1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 

K1c (MPa-m A 0.5) 



Figure F-5. Two Parameter Fracture Model, Strain Rate = 3 /s 
Top: Curve Fit CTOD -vs- K Ic Data Groups 1-4 
Middle: Curve Fit CTOD -vs- K Ic Data Groups 5-8 
Bottom: Solution by Error Minimization 



137 



Coefficient 


76 mm Diam Holed-Nctehed Data 


102 mm Diam Holed-Notched Data 






Group 1 


Group 2 


Group 3 


Group 4 


Group 5 


Group 6 


Group 7 


Group 8 




a 


-0.0820034 


-0.205815 


-0 849777 


-1 23582 


-0.103279 


-0.269247 


-1.20206 


-1.87774 




6 


0.0862426 


0.273837 


1.26902 


2.39375 


0.0995846 


0.3162 


1.57833 


2.9772 






c 


-0.02592 


-0.119816 


-0.695503 


-1.71788 


-0.0274406 


-0.122118 


-0.760564 


-1.74893 






d 


0.00335328 


0.0190079 


0.168693 


0.541469 


0.00325475 


0.0171 


0.162196 


0.451235 






e 








-0.0150436 


-0.0625284 








-0.0127176 


-0.0426537 






K1 


CTOD Grp 1 


CTOD Grp 2 


CTOD Gp 3 


CTOD Grp 4 


CTOD Grp 5 


CTOD Grp 6 


CT0DGp7 


CTOD Grp 8 


CTOD-(avg) 


Minimized 


<MPa-m»0.5) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


Error, (mm) A 2 


1.9700 


0.012939 


0.013972 


0.014158 


0010915 


0.011292 


0.010475 


0.004082 


0.007358 


0.010648889 


3.1219E-05 


1.9712 


0.012966 


0.014000 


0.014197 


0.010936 


0.011327 


0.010516 


0.004179 


0.007400 


01069032 


3.0876E-05 


1.9724 


0.012994 


0.014028 


0.014237 


0.010957 


0.011363 


0.010557 


0.004276 


0.007443 


0.010731665 


3.0537E-05 


1.9736 


0.013022 


0.014055 


0.014275 


0.010978 


0.011398 


0.010598 


0004373 


0.007485 


0.010772924 


3.0202E-05 


1.9748 


0.013050 


0.014083 


0.014314 


0.010999 


0.011433 


0.010639 


0.004469 


0.007527 


0.010814099 


2.9871 E-05 


1.9760 


0.013077 


0.014111 


0.014353 


0.011020 


0.011468 


0.010679 


0.004565 


0.007569 


0.010855189 


2.9545E-05 


1.9772 


0.013105 


0.014138 


0.014391 


0.011041 


0.011503 


0.010720 


0.004661 


0.007610 


0.010896196 


2.9222E-05 


1.9784 


0.013133 


0.014166 


0.014430 


0.011062 


0.011538 


0.010760 


0.004757 


0.007651 


0.010937119 


2.8903E-05 


1.9796 


0.013160 


0.014193 


0.014468 


0.011083 


0.011573 


0.010801 


0.004852 


0007693 


0.010977958 


2.8587E-05 


1.9808 


0.013188 


0.014221 


0.014506 


0.011105 


0.011608 


0.010841 


0.004947 


0.007733 


0.011018716 


2.8276E-05 


1.9820 


0.013216 


0.014248 


0.014544 


0.011126 


0.011643 


0.010882 


0.005042 


0.007774 


0.01105939 


2.7968E-05 


1.9832 


0.013243 


0.014276 


0.014582 


0.011148 


0.011678 


0.010922 


0.005136 


0.007815 


0.011099983 


2.7665E-05 


1.9844 


0.013271 


0.014303 


0.014620 


0.011169 


0.011713 


0.010963 


0.005230 


0.007855 


0.011140494 


2.7365E-05 


1.9856 


0.013298 


0.014331 


0.014657 


0.011191 


0.011748 


0.011003 


0005324 


0007895 


0.011180924 


2.7068E-05 


19868 


0.013326 


0.014358 


0.014694 


0.011213 


0.011783 


0.011043 


0.005418 


0.007935 


0.011221274 


2.6776E-05 


1.9880 


0.013354 


0.014385 


0.014732 


0.011234 


0.011818 


0.011083 


0.005511 


0.007975 


0.011261543 


2.6487E-05 


1.9892 


0.013381 


0.014413 


0.014769 


0.011256 


0.011853 


0.011123 


0.005605 


0.008014 


0.011301732 


2.6201 E-05 


1.9904 


0.013409 


0.014440 


0.014806 


0.011278 


0.011888 


0.011163 


0.005697 


0.008053 


0.011341842 


2.591 9E-05 


1.9916 


0.013436 


0.014467 


0.014843 


0.011301 


0.011923 


0.011203 


0.005790 


0.008093 


0.011381873 


2.5641 E-05 


1.9928 


0.013464 


0.014495 


0.014880 


0.011323 


0.011957 


0.011243 


0.005882 


0.008131 


0.011421824 


2.5366E-05 


1.9940 


0.013491 


0.014522 


0.014916 


0.011345 


0.011992 


0.011283 


0.005974 


0.008170 


0.011461698 


2.5095E-05 


19952 


0.013518 


0.014549 


0.014953 


0.011367 


0.012027 


0.011322 


0.006066 


0.008209 


0.011501494 


2.4827E-05 


1.9964 


0.013546 


0.014577 


0.014989 


0.011390 


0.012062 


0.011362 


0.006158 


0.008247 


0.011541212 


24563E-05 


1.9076 


0.013573 


0.014604 


0.015025 


0.011412 


0.012096 


0.011402 


0.006249 


0008285 


0.011580853 


2.4302E-05 


1.9988 


0.013601 


0.014631 


0.015061 


0.011435 


0.012131 


0.011441 


0.006340 


0.008323 


0.011620417 


2.4045E-05 


2.0000 


0.013628 


0.014658 


0.015097 


0.011458 


0.012166 


0.011481 


0006430 


0008361 


0.011659905 


2.3790E-05 


2.0012 


0.013655 


0.014685 


0.015133 


0.011480 


0.012200 


0.011520 


0006521 


0.008398 


0.011699317 


2.3540E-05 


2.0024 


0.013683 


0.014713 


0.015169 


0.011503 


0.012235 


0.011560 


0006611 


0.008436 


0.011738653 


2.3292E-05 



2.5496 
2.5508 
2.5520 
25532 
2.5544 
2.5556 
2.5568 



0.024965 
0024988 
0.025011 
0.025035 
0.025058 
0.025081 
0.025105 



0.028530 
0.028570 
0.028610 
0.028651 
0.028691 
0.028732 
0.028772 



0.024791 
0024808 
0.024828 
0024844 
0.024863 
0.024881 
0.024899 



0.032152 
0.032209 
0.032267 
0.032324 
0.032382 
0.032439 
0.032497 



0.026188 
0.026216 
0.026244 
0.026271 
0.026299 
0.026327 
0.026355 



0.026521 
0.026554 
0.026586 
0.026619 
0.026651 
0.026684 
0.026716 



0.028807 
0.028831 
0.028855 
0.028879 
0.028902 
0.028926 
0.028950 



0.020282 


0.026529302 


0.020319 


0.02656189 


0.020356 


0.026594494 


0.020394 


026627115 


0.020432 


0.026659754 


0.020469 


0.026692409 


0.020507 


0.026725082 



2.0294E-05 
2.0299E-05 
2.0303E-05 
2.0308E-05 
2.031 2E-05 
2.031 6E-05 
2.0321 E-05 



Figure F-6. Results of Curve Fit CTOD -vs- K Ic Data and Solution by Error 
Minimization, Strain Rate ■ 4 /s 



E 
E. 

u 

Q 

o 



E 
E, 

o 

Q 

o 

r- 
o 



0.050 

0.045 -3 

0.040 

0.035 -_ 

0.030 

0.025 

0.020 

0.015 -a 

0.010 

0.005 

0.000 



138 



Group * and Speci men Type 

Group 1 , a/R=0.08, r/R=0.25 

^ — Group 2, a/R=0.08, r/R=0.45 
-A — Group3,a/R=0.12, r/R=0.25 
— | Group 4,a/R=0.12, r/R=0.45 




I II I I I {I I | I I I I I I I I I I I I I I | I I I I | I I I I I I I I I I I I I I I I I I I 
1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 



K1c(MPa-m A 0.5) 



0.060 
0.055 -I 
0.050 
0.045 -| 
0.040 
0.035 
0.030 
0.025 -§ 
0.020 4 
0.015 -§ 
0.010 4 
0.005 
0.000 



Group # and Specimen Type 

Group 5, a/R=0.08, r/R=0.25 

- H — Group 6, a/R=0.08, r/R=0.45 
-+ — Group 7, a/R=0.12, r/R=0.25 
-# — Group8,a/R=0.12,r/R=0.45 




1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 
K1c(MPa-m*0.6) 



3.O0E-O5 

J 3 " 2.75E-05 

| 2.50E-05 

C 2.25E-05 
O 

^ 2.00E-05 3 
"S 1.75E-05 : 

a 

E 1.50E-O5 
5 1.25E-05J 



1.00E-05 I i i i I I i i i I I I i i i I I i I i i i i i i i i i i i i i i i i 

1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 

K1c (MPa-m A 0.5) 



Figure F-7. Two Parameter Fracture Model, Strain Rate = 4 /s 
Top: Curve Fit CTOD -vs- K lc Data Groups 1-4 
Middle: Curve Fit CTOD -vs- K, c Data Groups 5-8 
Bottom: Solution by Error Minimization 













139 












Coefficient 


76 mm Diam Holed-Notched Data 


102 mm Diam Holed Notched Data 






Group 1 


Group 2 


Group 3 


Group 4 


Group 5 


Group 6 


Group 7 


Group 8 


a 


-0.0888843 


-0.224605 


-0.948978 


-142205 


-0.112746 


-0.312844 


-1.32636 


-2.1439 




1 


0.0862427 


0.273838 


1.26902 


2.39375 


0.0995846 


0316201 


1.57833 


2.9772 








c 


-0.0239135 


-0.109792 


-0.622798 


-1.49291 


-0.0251367 


-0.1051 


-0 68929 


-1.5318 








d 


0.00285421 


0.0159607 


0.135268 


0.408936 


0.00273115 


0.0126661 


0.133221 


0.346149 








a 








-0.0108019 


-0.0410394 








-0 00946682 


-0.0286581 




Minimized 






CTOOGrpI CTODGrp2 


CTOOGp3 


CTOD Grp 4 


CTOD Grp 5 


CTOD Grp 6 


CTODGp7 








K1 


CTOD Grp 8 


CTOD-(avg) 




<MP>-m"0.5) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


Error, (mm)*2 


2.2500 


0.016611 


0.017511 


0.017348 


0.012272 


0.015174 


0.010814 


0.010198 


0.008440 


0.013546159 


31867E-05 




2.2515 


0.016643 


0.017543 


0.017390 


0.012296 


0.015215 


0.010866 


0.010297 


0.008492 


0.013592667 


3.1614E-05 




2.2529 


0.016675 


0.017575 


0.017431 


0.012320 


0.015255 


0.010917 


0.010396 


0.008543 


0.013639074 


31365E-05 




2.2544 


0.016706 


0.017607 


0.017472 


0.012345 


0.015296 


0.010969 


0.010495 


0.008593 


0.013685381 


3.1121E-05 




2.2558 


0.016738 


0.017639 


0.017513 


0012369 


0.015336 


0.011020 


0.010593 


0.008644 


0.013731588 


3.0880E-05 




2.2573 


0.016770 


0.017672 


0.017554 


0.012393 


0.015377 


0.011071 


0.010691 


0.008694 


0.013777695 


3.0643E-05 




22587 


0.016802 


0.017704 


017595 


0.012418 


0.015417 


0.011123 


0.010788 


0.008744 


0.013823703 


3.0409E-05 




2.2602 


0.016834 


0.017736 


0.017635 


0012442 


0015457 


0011174 


010885 


0008794 


0.013869613 


3 0180E-05 




2.2616 


0.016865 


0.017768 


0.017676 


0.012467 


0.015498 


0.011225 


0.010982 


0008843 


0.013915426 


2.9954E-05 




2.2631 


0.016897 


0.017800 


0.017716 


0.012491 


0.015538 


0.011276 


0.011079 


0008893 


0.013961141 


2.9732E-05 




2.2645 


0.016929 


0.017832 


0.017756 


0.012516 


0.015578 


0.011327 


0.011175 


0.008941 


0.014006759 


2.951 4E-05 




2.2660 


0.016960 


0.017864 


0.017796 


0.012541 


0.015616 


0.011377 


0.011271 


0.008990 


0.014052282 


2.9299E-05 




2.2674 


0.016992 


0.017896 


0.017836 


0012566 


0.015659 


0.011428 


0.011367 


0009038 


0.014097709 


29088E-05 




2.2689 


0.017024 


0.017928 


0.017875 


0.012591 


0.015699 


0.011479 


0.011462 


0.009087 


0014143041 


2.8880E-05 




2.2703 


0.017055 


0.017960 


0.017915 


0.012616 


0.015739 


0.011529 


0.011557 


0.009135 


0.014188279 


2.8676E-05 




2.2718 


0.017087 


0.017992 


0.017954 


0.012642 


0.015779 


0.011580 


0.011651 


0.009182 


0.014233423 


2.8475E-05 




2.2732 


0.017118 


0.018024 


0.017993 


0012667 


0.015819 


0.011630 


0.011746 


0.009230 


0014278473 


2.8277E-05 




2.2747 


0.017150 


0.01 8057 


0.018032 


0.012693 


0.015859 


0011681 


0011840 


0.009277 


0.014323431 


2.8083E-05 




2.2761 


0.017181 


0.018089 


0.018071 


0.012718 


0.015899 


0.011731 


0.011934 


0.009324 


014368297 


2.7893E-05 




2.2776 


0.017213 


0.018121 


0.018110 


0.012744 


0.015939 


0.011781 


0.012027 


0009370 


0.01441J071 


27706E-05 




2.2790 


0.017245 


0.018153 


0.018148 


0.012770 


0.015979 


0.011831 


0.012120 


0.009417 


0014457754 


27522E-05 




2.2805 


0.017276 


0018185 


0.018187 


0.012796 


0.016019 


0.011881 


0.012213 


0.009463 


0.01 4502346 


2.7341 E-05 




2.2819 


0.017307 


0.018217 


0.018225 


0.012822 


0.016059 


0.011931 


0.012305 


0.009509 


014546849 


2.7164E-05 




2.2834 


0.017339 


0.018249 


0.018263 


0.012848 


0.016099 


0.011981 


0.012398 


0009554 


0.014591262 


2.6989E-05 




2.2848 


0.017370 


0.018281 


0.018301 


0012874 


0.016139 


0.012031 


0.012489 


0.009600 


014635586 


26818E-05 




2.2863 


0.017402 


0.018313 


018339 


0.012900 


0.016179 


0.012080 


0.012581 


0.009645 


014679821 


26650E-05 




2.2877 


0.017433 


0.018345 


0.018377 


0.012927 


0.016219 


0.012130 


0.012672 


0.009690 


0.014723969 


2.6485E-05 




2.2892 


0.017464 


0.018377 


0.018414 


0.012953 


0.016258 


0.012179 


0.012763 


0.009735 


0.014768029 


2 6323E-05 




2.9504 


0.030706 


0.0375 1 e 


0029322 


0.037781 


0032399 


0.030494 


0.034317 


0.024451 


0032123371 


3.9790E-05 




2.9518 


0.030734 


0.037578 


0.029347 


0.037849 


0.032432 


0.030533 


0.034343 


0.024500 


0032164605 


3.9S55E-05 




2.9533 


0.030763 


0.037641 


0.029371 


0.037918 


0.032465 


0.030572 


0.034368 


0024549 


0032205873 


3.9922E-05 




2.9547 


0.030791 


0.037703 


0.029396 


0.037987 


0.032496 


0.030611 


0.034393 


0.024599 


0032247174 


3.9989E-05 




2.9562 


0.030820 


0.037765 


0.029420 


0.038056 


0.032530 


0.030650 


0.034418 


0.024648 


0.032288507 


4.0056E-05 




2.9576 


0.030848 


0.037828 


0.029445 


0.038124 


0.032563 


0.030689 


0.034444 


0024698 


0.032329874 


4.0124E-05 




2.9591 


0.030877 


0.037891 


0.029469 


0.038193 


0.032596 


0.030728 


0.034469 


0.024748 


032371275 


4.0193E-05 


] 


^igure F-8 


. Results of Curve Fit CTOD -vs- K Jc Data and Solution by Erro 


J- 








Minimization, Strain Rate = 5 /s 




( 



140 



E 
E, 
o 

Q 

O 

l- 

o 



0.055 
0.050 
0.045 
0.040 
0.035 
0.030 
0.025 
0.020 
0.015 
0.010 
0.005 
0.000 



Group # art specimen Type 

Group 1 , a/R=0.08, r/R=0.25 

— Group 2, a/R=0.08, r/R=0.45 

-£s — Group 3, a/R=0.12, r/R=0.25 

— | Group4,a/R=0.12, r/R=0.45 




I I I I I l^l I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I M I I I I I I I I I I I 

1.60 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 
K1c(MPa-m A 0.6) 



0.070 

0.060 

_ 0.050 - 

E 

.§. 0.040 

a 

§ 0.030 



o 



0.020 



0.010 



0.000 



Group # and Speci men Type 

Group 5, a/R=0.08, r/R=0.25 

- R — Group 6, a/R=0.08. r/R=0.4S 
-+ — Group7.a/R=0.12. r/R=0.25 
-^ — Group 8, a/R=0.12, r/R=0.45 




I I I I | #l I I | I I I I | I I I I | I I I I | I II I | II I I I I I I I I M I I I I II I ] I I I I I I I I I I I I I I I 
1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 



K1c (MPa-m A 0.6) 



< 

E 
E 



o 

fc 

LU 

■o 

0) 
N 



3.40E-05 
3.20E-05 | 
3.00E-05 | 
2.80E-05 
2.60E-05 | 

2.40E-05 ;; 

2.20E-05 !■ 



2.00E-05 




-i — i — t — i — t 



I — M r— I — I — I — I — I — I — I- 



2.20 



2.30 



2.40 2.50 2.60 2.70 

K1c(MPa-m A 0.5) 



2.80 



2.90 



Figure F-9. Two Parameter Fracture Model, Strain Rate = 5 /s 
Top: Curve Fit CTOD -vs- K k Data Groups 1-4 
Middle: Curve Fit CTOD -vs- K ]c Data Groups 5-8 
Bottom: Solution by Error Minimization 



141 



Coefficient 



76 mm Diam. Holed-Notched Data 



Group 1 



Group 2 



Group 3 



Group 4 



-0.0957648 -0243394 

0.0862425 0773838 

-0.0221952 -0.101317 

00245878 0.0135915 



-1.04818 -180828 

1.26903 2 39375 

-0.563857 -132004 

0.110875 0.319716 



-0.00801606 -00283702 



102 mm Diam Holed-Notched Data 



Group 5 



Group 6 



Group 7 



Group 8 



-0.122212 -0.356441 

0.0995846 0.316201 

-0.0231896 -0.0922453 

0.00232443 0.00975719 



-1.45065 -2.41007 

1.57834 2.9772 

-0.630231 -1.36263 

0.11137 0.273914 



-0 00723595 -0.0201732 



K1 


CTOD Grp 1 


CTOD Grp 2 


CTODGp3 


CTOD Grp 4 


CTOD Grp 5 


CTOD Grp 6 


CTODGp7 


CTOD Grp 8 


CTOD-(avg) 


Minimized 


<MPa-mM).5) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


Error, (mm) A 2 


2.5300 


0.020178 


0.021001 


0.020384 


0.013683 


0.018945 


0.011105 


0015593 


0.009500 


0.016296102 


4.8203E-05 


2.5315 


0.020208 


0.021033 


0.02O421 


013686 


0.018984 


0.011159 


0015679 


0.009551 


0016340185 


4.8006E-05 


2.5329 


0.020239 


0.021065 


0.020458 


0.013709 


0.019023 


0.011212 


0.015765 


0.009602 


0.016384185 


4.781 2E-05 


2.5344 


0.020270 


0.021097 


0.020495 


013733 


0.019062 


0.011265 


0.015850 


0009653 


0.016428101 


4.7621 E-05 


2.5353 


0.020300 


0.021129 


0.020532 


0.013756 


0.019101 


011318 


0.015935 


0009704 


0.016471935 


4.7432E-05 


2.5373 


0.020331 


0.021161 


0.020569 


0.013780 


0.019140 


0.011371 


016020 


0.009754 


0016515686 


4.7247E-05 


2.5387 


0.020361 


0.021194 


0.020606 


0013803 


0.019179 


0.011424 


0016104 


0009804 


0016559356 


4.7064E-05 


2.5402 


0020392 


0.021226 


0.020642 


0.013827 


0.019217 


0.011477 


0.016188 


0.009854 


016602944 


4.6884E-05 


2.5416 


0.020423 


021258 


0.020679 


0.013851 


0.019256 


0.011530 


0.016272 


0009903 


0.016646452 


4.6707E-O5 


2.5431 


0.020453 


0.021290 


0.020715 


0.013875 


0.019295 


0.011582 


0.016356 


0.009953 


0016689879 


4.6533E-05 


2.5445 


0.020484 


0.021323 


0.020751 


0.013899 


0019334 


0.011635 


0.016439 


0.010002 


0016733227 


4.6362E-05 


2.5460 


0.020514 


0.021355 


0.020787 


0.013923 


0.019373 


0.011688 


0.016523 


0.010061 


0.016776495 


4.6193E-05 


2.5474 


0.020545 


0.021387 


0.020823 


013947 


0.019411 


0.011740 


0.016606 


0010099 


0.016819683 


4.6O27E-05 


2.5489 


0.020575 


0.021419 


0.020859 


0.013971 


0.019450 


0.011793 


0.016688 


0.010148 


0.016862794 


4.5864E-05 


2.5503 


0.020606 


0.021452 


0.020895 


0.013995 


0.019489 


0.011845 


0.016771 


0.010196 


0.016905826 


4.5703E-05 


2.5518 


0.020636 


021484 


0.020930 


0.014019 


0.019527 


0.011897 


0.016853 


0010244 


0.01694878 


4.5545E-05 


2.5532 


0.020666 


0.021516 


0.020966 


0.014044 


0.019566 


0.011949 


0.016935 


0.010291 


016991657 


45389E-05 


2.5547 


0.020697 


0.021548 


0.021001 


0.014068 


0.019604 


0,012002 


0.017016 


0.010339 


0.017034457 


4.5237E-05 


2.5561 


0.020727 


0.021581 


0.021036 


0.014093 


0.019643 


0.012054 


0.017098 


0.010386 


0.017077181 


4.5086E-05 


2.5576 


0.020758 


0.021613 


0.021071 


0014118 


0.019682 


0.012106 


0.017179 


0.010433 


0.017119829 


4.4938E-05 


2.5590 


0.02C788 


021845 


0.021106 


0.014142 


0019720 


0.012157 


0017260 


0.010480 


0017162401 


4.4793E-05 


2.5605 


0.020818 


0.021678 


0.021141 


0.014167 


0.019759 


0.012209 


0017340 


010526 


0.01720489a 


4 4650E-05 


2.5619 


0.020849 


0.021710 


0.021176 


0.014192 


0.019797 


0.012261 


0.017421 


0010573 


017247321 


4.451 OE-05 


2.5634 


0.020879 


0.021743 


0.021210 


0.014217 


0019836 


0.012313 


0.017501 


0.010619 


0.01728967 


4.4372E-05 


2.5648 


0.020909 


0.021775 


0.021245 


0.014242 


0.019874 


0.012364 


0.017581 


0.010665 


0017331944 


4.4236E-05 


2.5683 


0.020940 


0.021807 


0.021279 


0.014267 


0.019912 


0012416 


0.017660 


0.010711 


0017374146 


4.4103E-05 


2.5677 


0.020970 


0.021840 


0.021314 


0.014292 


0.019951 


0.012467 


0017740 


0.010756 


0.017416274 


4.3972E-05 


25692 


0.021000 


0.021872 


0.021348 


0014318 


0.019989 


0.012519 


017819 


0.010801 


0.01745833 


4.3843E-05 



3.2304 


0.034102 


0.042098 


0.031913 


0.037614 


0.035849 


0.031312 


0.037633 


0.024781 


0.03441259 


6.7346E-05 


3.2318 


0.034130 


0.042163 


0.031937 


0.037684 


0.035881 


0.031349 


0.037659 


0.024822 


0.034453194 


6.7487E-05 


3.2333 


0.034159 


0.042228 


0.031960 


0.037755 


0.035914 


0.031386 


0.037684 


0.024864 


0.034493834 


6 7629E-05 


3.2347 


0.034188 


0.0422S3 


0.031984 


0.037826 


0.035947 


0.031423 


0.037709 


0.024905 


0.03453451 


6.7771 E-05 


3.2362 


0.034217 


0.042359 


0.032008 


0.037897 


0.035979 


0.031481 


0.037734 


0.024947 


0034575224 


6.791 5E-05 


3.2376 


0.034246 


0.042424 


0.032031 


0.037968 


0.036012 


0.031498 


0037760 


0.024989 


0.034615973 


6.8059E-05 


32391 


0.034274 


0.042490 


0.032055 


0038039 


0.036045 


0.031535 


0.037785 


0.025031 


0.03465676 


6.8204E-05 







Figure F-10. Results of Curve Fit CTOD -vs- K Ic Data and Solution by Error 

Minimization, Strain Rate = 6 /s 



142 



E 

E. 

I 

O 

& 



E 
E 

u 
o 
o 



055 
050 
045 
040 
035 
030 
025 
020 
015 
010 
005 
000 



Group # and Specimen Type 

Group 1 , a/R=0.08, r/R=0.25 

# — Group 2, a/R=0.08. r/R=0.45 
-A — Group 3, a/R=0.12, r/R=0.25 
- Group 4, a/R=0.12, r/R=0.45 




| i &i i | i I I I | I I I I | I I I I | I I I I | I I I I | i i I I | I I I I | I I I I i I I 



1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 

K1c(MPa-m A 0.5) 



i i | m i i 
4.00 4.2 



080 

070 

060 

050 

040 

030^ 

020 4 

010- 



000 



Group # and Specimen Type 

Group 5, a/R=0.08, r/R=0.25 

□ Group 6, a/R=0.08, r/R=0.45 
-♦— Group 7, a/R=0.12, r/R=0.25 
Hft — Group 8, a/R=0.12, r/R=0.45 




| ''♦' | M I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | M I I I I I I I I 

2.00 2.20 240 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4 20 4 40 46 

K1c(MPa-m A 0.5) 



6.00E-05 




3.50E-05 



K1c (MPa-m A 0.6) 



3.10 



3.20 



Figure F-l 1 . Two Parameter Fracture Model, Strain Rate = 6 /s 
Top: Curve Fit CTOD -vs- K Ie Data Groups 1-4 
Middle: Curve Fit CTOD -vs- K lc Data Groups 5-8 
Bottom: Solution by Error Minimization 













143 












Coefficient 


76 mm Diam. Hoted-Notctied Data 


102 mm Diam Holed- Notched Data 






Group 1 


Group 2 


Group 3 


Group 4 


Group 5 


Group 6 


Group 7 


Group 8 


a 


■0.102646 


■0 262185 


-1.14738 


-1.7945 


-0.131678 


-0 400037 


-1 57495 


-267623 




1 


0.0862428 


0.273838 


1.26903 


2.39375 


0.0995846 


0.316201 


1.57833 


2.9772 








• 


■0.0207075 


-0.0940558 


-0.515106 


-1.18305 


-0.0215225 


-0.0821922 


-0.580492 


-1.22711 








d 


0.00214019 


0.0117132 


0.0925318 


0.256801 


0.00200224 


0.0074636 


0.0944848 


222139 








e 








-0.00611146 


-0 0204226 








-0 0056544 


-0.014733 




Minimized 






CTOO Grp 1 


CTOD Grp 2 


CTOD Gp 3 


CTOD Grp 4 


CTOD Grp 5 


CTOD Grp 6 


CTODGp7 








K1 


CTOD Grp 8 


CTOD-(avg) 




(MPwn"0.6) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


(mm) 


Error, (mm) A 2 


2.8100 


0.023674 


0.024519 


0023325 


0.015051 


0.022637 


0005093 


0.020424 


0.010569 


0.018161556 


1.2512E-04 




Z8114 


0.023702 


0.024549 


0.023357 


0.015072 


0.022672 


0.005135 


0.020495 


0.010617 


018199892 


1 2498E-04 




2.8127 


0.023730 


0.024580 


0.023389 


0.015094 


0.022707 


0.005176 


0.020565 


0.010664 


018238167 


1.2484E-04 




2.8141 


0.023758 


0.024611 


0.023421 


0.015115 


0.022742 


0.005218 


0020635 


0.010711 


0.018276379 


1.2471E-04 




2.8154 


0.023785 


0.024642 


0.023452 


0.015136 


0.022777 


0.005260 


0.020705 


0.010758 


0.01831453 


1.2457E-04 




2.8168 


0.023813 


0.024872 


0023484 


0.015157 


0.022812 


0.005302 


0.020775 


0.010805 


0.018352621 


1.2444E-04 




2.8181 


0.023841 


0.024703 


0.023515 


0.015179 


0.022847 


0.005343 


0.020845 


0.010852 


0.01839065 


1.2431E-04 




2.8195 


0.023868 


0.024734 


0.023547 


0.015200 


0.022883 


0.005385 


0.020914 


0.010898 


0.018428619 


1.2419E-04 




2.8208 


0.023896 


0.024765 


0.023578 


0.015222 


0.022918 


0.005426 


0.020983 


0010945 


018466528 


1.2406E-04 




2.8222 


0.023924 


0.024795 


0.023609 


0.015243 


0.022953 


0.005467 


0.021052 


0.010991 


0.018504377 


1.2394E-04 




2.8235 


0.023952 


0.024826 


0.023640 


0.015265 


0.022988 


0.005509 


0.021121 


0.011037 


018542166 


1.2382E-04 




2.8249 


0.023979 


0.024857 


0.023671 


0.015287 


0.023023 


0.005550 


0.021190 


0.011083 


018579896 


1.2370E-04 




2.8262 


0.024007 


0.024888 


0.023702 


0.015308 


0.023058 


0.005591 


0.021258 


0.011128 


0.018617567 


1.2356E-04 




2.8276 


0.024034 


0.024919 


0.023733 


0.015330 


0.023093 


0.005632 


0.021326 


0.011174 


0.01865518 


1.2347E-04 




2.8289 


0.024062 


0.024950 


0.023764 


0.015352 


0.023128 


0.005673 


0.021395 


0.011219 


0.018692734 


12335E-04 




2.8303 


0.024090 


0.024981 


0.023794 


0.015374 


0.023162 


0.005714 


0.021463 


011264 


0018730229 


1.2324E-04 




2.8316 


0.024117 


0.025012 


0.023825 


0.015396 


0.023197 


0.005755 


0.021530 


0.011309 


0.018767667 


1.2313E-04 




2.8330 


0.024145 


0.025043 


0.023856 


0.015418 


0.023232 


0.005796 


0.021598 


0.011353 


0018805048 


1.2303E-04 




2.8343 


0.024173 


0.025074 


0.023886 


0.015440 


0.023267 


0.005837 


0.021865 


0.011398 


0.018842371 


1.2292E-04 




2.8357 


0.024200 


0.025105 


0.023916 


0.015462 


0.023302 


0.005877 


0.021732 


0.011442 


0.018879638 


1.2282E-04 




2.8370 


0.024228 


0.025136 


0.023947 


0015484 


0023337 


0.005918 


0.021799 


0.011486 


0.018916847 


1.2271E-04 




2.8384 


0.024255 


0.025167 


0.023977 


0.015507 


0.023372 


0.005958 


0.021866 


0.011530 


018954001 


1.2261E-04 




2.8397 


0.024283 


0.025198 


0.024007 


015529 


0.023407 


0.005999 


0.021933 


0.011574 


0.018991098 


1.2251E-04 




2.8411 


0.024310 


0.025229 


0.024037 


0.015551 


0.023441 


0.006039 


0.021999 


0.011618 


0019028139 


12242E-04 




2.8424 


0.024338 


0.025260 


0.024067 


0.015574 


0.023476 


0.006030 


0022066 


0.011661 


019065125 


1.2232E-04 




28438 


0.024365 


0.025291 


0.024097 


0.015596 


0.023511 


0.006120 


0.022132 


0011705 


019102056 


1.2223E-04 




34620 


0.036543 


0044567 


0.033790 


0.035161 


0.038208 


0.019233 


0.040031 


0.024320 


0.033981612 


1.7617E-04 




3.4634 


0.036570 


0.044626 


0.033811 


0.035223 


0.038239 


0.019254 


0.040055 


0024350 


0.034015991 


1.7649E-04 




3.4647 


0.036597 


0.044685 


0.033831 


0035286 


0038269 


0.019275 


0.040079 


0.024381 


0034050396 


1.7680E-04 




3.4661 


0.036623 


0.044744 


0.033852 


0035349 


0.038300 


0.019296 


0.040103 


0.024412 


0.034084829 


1.7712E-04 




3.4674 


0.036650 


0.044804 


0.033872 


0.035411 


0.038330 


0.019317 


0.040127 


0.024443 


0.03411929 


1.7744E-04 




3.4688 


0.036677 


0.044863 


0.033893 


0.035474 


0.038361 


0.019337 


0.040151 


0.024474 


0.034153779 


1.7777E-04 




3.4701 


0.036704 


0.044923 


0.033914 


0035537 


0.038391 


0.019358 


0.040175 


0.024505 


0.034188296 


1.7809E-04 


F 


'igure F-12. Results of Curve Fit CTOD - 


vs- K Ic Data and Solution by Err< 


jr 








Minimization, Strain Rate - 7 It 







144 



E 
E. 

u 
a 
o 



0.065 
0.060 
0.055 
0.050 
0.045 
0.040 
0.035 
0.030 
0.025 
0.020 
0.015 
0.010 
0.005 
0.000 





GrouD # and Specimen Tvpe 




-3 


— — Group 2, a/R=0.08, r/R=0.45 
— A — Group3,a/R=0.12, r/R=0.25 




- 






4 




Rale - 7/s 




I I I I I M I M I I I I I I I I I | I I I I I I I I I M I I I | 


l l l l 1 l l l l 1 l l l l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 II 1 1 1 1 1 1 



2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 
K1c(MPa-m A 0.6) 



E 

E, 

o 
o 
o 

h- 

o 



0.090 
0.080 
0.070 
0.060 
0.050 
0.040 
0.030 
0.020 
0.010 
0.000 



Group » and Specimen Type 

Group 5, a/R=0.08, r/R=0.25 

- R — Group 6, a/R=0.08, r/R=0.45 
-^ — Group7,a/R=0.12, r/R=0.25 
-# — Group8,a/R=0.12,r/R=0.45 




1 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 



2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 
K1c(MPa-m A 0.6) 



1 1 1 1 1 1 1 1 
5.20 5.40 




.10E-04 -I— i — I I i i i — i— i— i — i — i — i — i— i — I I I i i i — I I I I | I I l — |— | — l | | 
2.80 2.90 3.00 3.10 3.20 3.30 3.40 

K1c (MPa-m A 0.5) 



3.50 



Figure F-13. Two Parameter Fracture Model, Strain Rate = 7 /s 
Top: Curve Fit CTOD -vs- K Ic Data Groups 1-4 
Middle: Curve Fit CTOD -vs- K lc Data Groups 5-8 
Bottom: Solution by Error Minimization 



APPENDIX G 
CALCULATIONS OF THE SIZE EFFECT METHOD 





Peak stress from regression j 
fit of experiment data I 




Geometry functions and 
their derivatives, 
Equations (9), (16), (17) 












i Regression values f< 
i Size Effect Method, 


)rthe 




\ 


Equation (10) '• 




Strain Rate ■ 2 /s \ 


u 




Ix. 




/ 


\ 




r~ 




"~ A 


\ 




Group # 


Specimen Type Notch On) D (in) (j n (MPa) CXO F(a/R) 


g(a/R 


9'(a/R) 


x=(g/g')D 


Y(1/MPa) A 2 




1 


r/R=0.08, aO/R=25 0.75 3 6.403 0.25 1.1892 


1125 0.4073 


0.01053 


0.006 




2 


r/R=0.08, a0/R=45 1.35 3 4.260 0.45 1.3146 


02475 0.9999 


000943 


0.006 




3 


r/R=0.12, a0/R=25 0.75 3 5.270 0.25 1.4364 


0.1642 0.2491 


0.02511 


0.015 




4 


r/R=0.12, aO/R=45 1.35 3 3.395 0.45 1.4296 


0.2927 1.1081 


0.01007 


0.008 




5 


r/R=0.08, a0/R=.25 1 4 5.936 0.25 1.1892 


0.1125 0.4073 


0.01404 


0.007 




6 


r/R=0.O8, aO/R=45 1.8 4 3.458 0.45 1.3146 


0.2475 0.9999 


0.01258 


0.008 




7 


r/R=0.12, a0/R=.25 1 4 5.376 0.25 1.4364 


0.1642 0.2491 


0.03348 


0.014 




8 


r/R=0.12, aO/R=45 1.8 4 3.684 0.45 1.4296 


0.2927 1.1081 


0.01342 


0.007 












Slope = 


0.3785 
0.002735 






i Linear Regression of the X and \ 


, ^r,^ . „ ; 






abM Parameters ;" 






lntercept= 












i Fracture Parameters from t 
| SEM model. Equations (1 


he Hneari7Rd 1 




Kn (MPa-m A 0.5) = 
~~ »• Ct (m)= 


1.625 
0.0072247 






3) and (12) ; 

















Figure G-l . Explanation of Spreadsheet Algorithm for the Size Effect Method 



146 



147 



Strain Rate 


1.00E-08 


per second 


















Group tt 


Specimen Type 


Notch On) 


D(in) 


On (MPa) 


ao 


F(a/R) g(a/R) 


g'(a/R) 


x=(g/g')D 


Y (1/MPa) A 2 


1 


r/R=0.08, aO/R=.25 


0.75 


3 


3.482 


0.25 


1.1892 0.1125 


0.4073 


0.01053 


0.021 


2 


r/R=0.08, aO/R=.45 


1.35 


3 


2.544 


0.45 


1.3146 0.2475 


0.9999 


0.00943 


0.016 


3 


r/R=0.12, aO/R=.25 


0.75 


3 


3.096 


0.25 


1.4364 0.1642 


0.2491 


0.02511 


0.042 


4 


r/R=0.12. aO/R=45 


1.35 


3 


2.213 


0.45 


1.4296 0.2927 


1.1081 


0.01007 


0.019 


5 


r/R=0.08, aO/R=.25 


1 


4 


3.261 


0.25 


1.1892 0.1125 


0.4073 


0.01404 


0.023 


6 


r/R=0.08, aO/R=.45 


1.8 


4 


2.268 


0.45 


1.3146 0.2475 


0.9999 


0.01258 


0.020 


7 


r/R=0.12. aO/R=.25 


1 


4 


2.792 


0.25 


1.4364 0.1642 


0.2491 


0.03348 


0.052 


8 


r/R=0.12, a0/R=.45 


1.8 


4 


2.179 


0.45 


1.4296 0.2927 


1.1081 


0.01342 


0.019 




















Slope = 


1.5262 




















Intercepts 


0.001927 






Ku (MPa-m A 0.5) = 


0.809 




















O (m)= 


0.0012625 



CM 



O 



>■ 



0.055 
0.050 - 
0.045 - 
0.040 
0.035 
0.030 - 
0.025 - 
0.020 



0.015 



Experiment Data from all 8 Cylinder Groups 
Linear Regression for Fracture Parameters 



T I i — I — l — i — i — I — r 



Coefficient of determination, R-squared = 0.978805 



n — i — r 



t i i I — i — i — i — r 



0.005 0.010 0.015 0.020 0.025 0.030 0.035 

X, (m) 



Figure G-2. Size Effect Method for Quasi-Static Strain Rate, using Groups 1-8 



148 



Strain Rate 


1.00E-O8 


per second 


















Group # 


Specimen Type 


Notch (in) 


D(in) 


an (MPa) 


010 


F(a/R) 


g(a/R) 


g'(a/R) 


x=(gVg')D 


Y (1/MPa) A 2 


1 


r/R=0.08, aO/R=25 


0.75 


3 


3.482 


0.25 


1.1892 


0.1125 


0.4073 


0.01053 


0.021 


2 


r/R=0.08, aO/R=. 45 


1.35 


3 


2.544 


0.45 


1.3146 


0.2475 


0.9999 


0.00943 


0.016 


3 


r/R=0.12, aO/R=25 


0.75 


3 


3.096 


0.25 


1.4364 


0.1642 


0.2491 


0.02511 


0.042 


4 


i7R=0.12,aOVR=45 


1.35 


3 


2.213 


0.45 


1.4296 


0.2927 


1.1081 


0.01007 


0.019 


5 


r/R=0.08, a0/R=25 


1 


4 


3.261 


0.25 


1.1892 


0.1125 


0.4073 


0.01404 


0.023 


6 


r/R=0.08, aO/R=45 


1.8 


4 


2.268 


0.45 


1.3146 


0.2475 


9999 


0.01258 


0020 


7 


r/R=0.12, aO/R=25 


1 


4 


2.792 


0.25 


1.4364 


0.1642 


0.2491 


0.03348 


0.052 


8 


r/R=0.12,aO/R=45 


1.8 


4 


2.179 


0.45 


1.4296 


0.2927 


1.1081 


0.01342 


0.019 


9 


r/R=0.12, aO/R=25 


1.5 


6 


2.972 


0.25 


1.1892 


0.1125 


0.4073 


0.02105 


0.028 


10 


r/R=0.12, aO/R=45 


27 


6 


1.951 


0.45 


1.4296 


0.2927 


1.1081 


0.02013 


0.024 




















Slope = 


1.4336 




















Intercept 


0.002048 






Kif (MPa-m A 0.5) = 


0.835 




















Cf (m)= 


0.0014284 



o 

CL 



> 



0.055 
0.050 
0.045 - 
0.040 - 
0.035 
0.030 - 
0.025 - 
0.020 - 



0.015 



Experiment Data from all 10 Cylinder Groups 
Linear Regression for Fracture Parameters 



Coefficient of determination, R-squared = 0.843063 
~i I I I | I i i I | — I — I — I — I — I — I — I — I — i — r 



~~ 1 I I I — | — I — I — i — I — I 

0.005 0.010 0.015 0.020 0.025 0.030 0.035 

X, (m) 



Figure G-3. Size Effect Method for Quasi-Static Strain Rate, using All 10 Groups 



149 



Strain Rate = 



Is 



Group* Specimen Type Notch (in) D (in) an (MPa) OtO F(a/R) BN") 9'(a/R) 



1 r/R=0.08, aOIR=25 0.75 3 

2 r/R=0.08,aO/R=.45 1.35 3 

3 r/R=0.12, aO/R=.25 0.75 3 

4 r/R=0.12, aO/R=.45 1.35 3 



5 r/R=0.08, aO/R=.25 1 4 

6 r/R=0.08, aO/R=.45 1.8 4 

7 r/R=0.12, aO/R= .25 1 4 

8 r/R=0.12, aO/R=.45 1.8 4 



6.403 0.25 1.1892 0.1125 0.4073 

4.260 0.45 1.3146 0.2475 0.9999 

5.270 0.25 1.4364 0.1642 0.2491 

3.395 0.45 1.4296 0.2927 1.1081 



5.936 0.25 1.1892 0.1125 0.4073 

3.458 0.45 1.3146 0.2475 0.9999 

5.376 0.25 1.4364 0.1642 0.2491 

3.684 0.45 1.4296 0.2927 1.1081 



X=(g/g-)D Y (1/MPa)*2 



0.01053 
0.00943 
0.02511 
0.01007 



0.01404 
0.01258 
0.03348 
0.01342 



Slope = 
lntercept= 



0.006 
0.006 
0.015 
0.008 



0.007 
0.008 
0.014 
0.007 



0.3785 
0.002735 



Kif (MPa-m A 0.5) = 1.625 

O (m)= 0.0072247 



CM 

< 

O 
Q_ 



>- 



0.016 



0.014 



0.012 - 



0.010 - 



0.008 



0.006 



0.004 



Experiment Data from all 8 Cylinder Groups 
Linear Regression for Fracture Parameters 



Coefficient of determination, R-squared = 0.846133 



T — I — I — I — | — I — I — I — I — | — I — I — I — I — | — I I i I | I I I I | i I i I | 

0.005 0.010 0.015 0.020 0.025 0.030 0.035 

X, (m) 

Figure G-4. Size Effect Method for Strain Rate = 2 /s, using Groups 1-8 



150 





Strain Rate = 


3 


/s 
















Group # 


Specimen Type 


Notch (in) 


D(in) 


an (MPa) 


ao 


F(a/R) 


g(a/R) 


g'(a/R) 


X=(g/g')D 


Y (1/MPa} A 2 


1 


r/R=0.08, aO/R=.25 


0.75 


3 


7.049 


0.25 


1.1892 


0.1125 


0.4073 


0.01053 


0.005 


2 


r/R=0.08, aO/R=.45 


1.35 


3 


4.736 


0.45 


1.3146 


0.2475 


0.9999 


0.00943 


0.005 


3 


r/R=0.12, aO/R=.25 


0.75 


3 


6.072 


0.25 


1.4364 


0.1642 


0.2491 


0.02511 


0.011 


4 


r/R=0.12, aO/R=.45 


1.35 


3 


4.127 


0.45 


1.4296 


0.2927 


1.1081 


0.01007 


0.005 


5 


r/R=0.08, a07R=.25 


1 


4 


6.602 


0.25 


1.1892 


0.1125 


0.4073 


0.01404 


0.006 


6 


r/R=0.08, aO/R=.45 


1.8 


4 


4.286 


0.45 


1.3146 


0.2475 


0.9999 


0.01258 


0.C06 


7 


r/R=0.12, aO/R=.25 


1 


4 


6.077 


0.25 


1.4364 


0.1642 


0.2491 


0.03348 


0.011 


8 


r/R=0.12, aO/R=.45 


1.8 


4 


4.413 


0.45 


1.4296 


0.2927 


1.1081 


0.01342 


0.005 




















Slope = 


0.3037 




















lntercept= 


0.001724 






Ku (MPa-m A 0.5) = 


1.815 




















a (m)= 


0.0056764 



< 
Q. 



>- 



0.012 



0.010 



0.008 



0.006 



0.004 



Experiment Data from all 8 Cylinder Groups 
Linear Regression for Fracture Parameters 



Coefficient of determination, R-squared = 0.901089 
1 r -1 r-| f— 1 l l r— i i i i I — i — i — | — | — | — | — | — | — | — | — | — r— | — | — 



0.005 0.010 0.015 0.020 0.025 0.030 0.035 

X, (m) 



Figure G-5. Size Effect Method for Strain Rate = 3 /s, using Groups 1-8 



151 





Strain Rate = 


4 


Is 
















Group # 


Specimen Type 


Notch (in) 


D(in) 


On (MPa) 


oto 


F(a/R) 


g(a/R) 


g'(a/R) 


X=(glg-)D 


Y (1/MPa) A 2 


1 


r/R=0.08, aO/R=.25 


0.75 


3 


7.695 


0.25 


1.1892 


0.1125 


0.4073 


0.01053 


0.004 


2 


r/R=0.08, aO/R=45 


1.35 


3 


5.212 


0.45 


1.3146 


0.2475 


0.9999 


0.00943 


0.004 


3 


r/R=0.12,aO/R=.25 


0.75 


3 


6.873 


0.25 


1.4364 


0.1642 


0.2491 


0.02511 


0.009 


4 


r/R=0.12,aO/R=.45 


1.35 


3 


4.860 


0.45 


1.4296 


0.2927 


1.1081 


0.01007 


0.004 


5 


r/R=0.08, aO/R=.25 


1 


4 


7.268 


0.25 


1.1892 


0.1125 


0.4073 


0.01404 


0.005 


6 


r/R=0.08, aO/R=45 


1.8 


4 


5.114 


0.45 


1.3146 


0.2475 


0.9999 


0.01258 


0.004 


7 


r/R=0.12, aO/R=.25 


1 


4 


6.778 


0.25 


1.4364 


0.1642 


0.2491 


0.03348 


0.009 


8 


r/R=0.12,aO/R=.45 


1.8 


4 


5.142 


0.45 


1.4296 


0.2927 


1.1081 


0.01342 


0.003 




















Slope = 


0.2465 




















lntercept= 


0.001199 






Kit (MPa-m A 0.5) = 


2.014 




















O (m)= 


0.0048655 



CN 
< 
D 

a. 



> 



0.010 -i 



0.008 



0.006 



0.004 - 



0.002 



Experiment Data from all 8 Cylinder Groups 
Linear Regression for Fracture Parameters 



Coefficient of determination, R-squared = 0.90336/ 



1 — I — I — I — | — r - r 
0.005 0.010 



~i | — i — i — i — i — | — i — i 

0.015 0.020 

X, (m) 



~i — | — i — i — r 
0.025 



1 I ' ' ' ' I 
0.030 0.035 



Figure G-6. Size Effect Method for Strain Rate = 4 /s, using Groups 1-8 



152 





Strain Rate ■ 


5 


/s 
















Group # 


Specimen Type 


Notch (in) 


D(in) 


an (MPa) 


ao 


F(a/R) g(a/R) 


g'(a/R) 


X=(g/g-)D 


Y (1/MPa) A 2 


1 


r/R=0.08, aO/R=.25 


0.75 


3 


8.340 


0.25 


1.1892 0.1125 


0.4073 


0.01053 


0.004 


2 


r/R=0.08, aO/R=.45 


1.35 


3 


5.688 


0.45 


1.3146 0.2475 


0.9999 


0.00943 


0.003 


3 


r/R=0.12,aO/R=25 


0.75 


3 


7.674 


0.25 


1.4364 0.1642 


0.2491 


0.02511 


0.007 


4 


r/R=0.12, aO/R=.45 


1.35 


3 


5.592 


0.45 


1.4296 0.2927 


1.1081 


0.01007 


0.003 


5 


r/R=0.08, aO/R=.25 


1 


4 


7.934 


0.25 


1.1892 0.1125 


0.4073 


0.01404 


0.004 


6 


r/R=0.08, aO/R=.45 


1.8 


4 


5.942 


0.45 


1.3146 0.2475 


0.9999 


0.01258 


0.003 


7 


r/R=0.12, aO/R=.25 


1 


4 


7.478 


0.25 


1.4364 0.1642 


0.2491 


0.03343 


0.007 


8 


r/R=0.12, aO/R=. 45 


1.8 


4 


5.870 


0.45 


1.4296 0.2927 


1.1081 


0.01342 


0.003 




















Slope = 


0.2031 




















lntercept= 


0.000895 






<if (MPa-m A 0.5) = 


2.219 




















Cf (m)= 


0.0044061 



CN 



D 



> 



0.008 -, 



0.006 - 



0.004 - 



0.002 



Experiment Data from all 8 Cylinder Groups 
Linear Regression for Fracture Parameters 



Coefficient of determination, R-squared = 0.8873 
1 I r-| l l |— I r— l : l — I — I — i — i — i — i — I — i — i — r— 



i — i — r 



0.005 0.010 0.015 0.020 0.025 0.030 035 

X, (m) 



Figure G-7. Size Effect Method for Strain Rate * 5 /s, using Groups 1-8 



153 





Strain Rate = 


6 


/s 
















Group # 


Specimen Type 


Notch (in) 


D(in) 


On (MPa) 


oto 


F(a/R) 


g(a/R) 


g'(a/R) 


X=(g/g , )D 


Y (1/MPa) A 2 


1 


r/R=0.08, aO/R=.25 


0.75 


3 


8.986 


0.25 


1.1892 


0.1125 


0.4073 


0.01053 


0.003 


2 


r/R=0.08, aO/R=.45 


1.35 


3 


6.164 


0.45 


1.3146 


0.2475 


0.9999 


0.00943 


C.003 


3 


r/R=0.12, aO/R=.25 


0.75 


3 


8.475 


0.25 


1.4364 


0.1642 


0.2491 


0.02511 


0.006 


4 


r/R=0.12, aO/R=.45 


1.35 


3 


6.324 


0.45 


1.4296 


0.2927 


1.1081 


0.01007 


0.002 


5 


r/R=0.08, aO/R=.25 


1 


4 


8.601 


0.25 


1.1892 


0.1125 


0.4073 


0.01404 


0.003 


6 


r/R=0.08, aO/R=.45 


1.8 


4 


6.770 


0.45 


1.3146 


0.2475 


0.9999 


0.01258 


0.002 


7 


r/R=0.12, aO/R=.25 


1 


4 


8.179 


0.25 


1.4364 


0.1642 


0.2491 


0.03348 


0.006 


8 


r/R=0.12,aO/R=.45 


1.8 


4 


6.599 


0.45 


1.4296 


0.2927 


1.1081 


0.01342 


0.002 




















Slope = 


0.1697 




















lntercept= 


0.000702 






Ku (MPa-m A 0.5) = 


2.427 




















© (m)= 


0.0041354 



CM 

< 

a. 

2 



>- 



0.008 



0.006 



0.004 - 



0.002 



Experiment Data from all 8 Cylinder Groups 
Linear Regression for Fracture Parameters 



Coefficient of determination, R-squared = 0.865603 



"i — i — i — i — i — i — r 



I I I I I I I I I I I I I I 1 T" 

0.005 0.010 0.015 0.020 0.025 0.030 

X, (m) 



0.035 



Figure G-8. Size Effect Method for Strain Rate = 6 /s, using Groups 1-8 



154 





Strain Rate ■ 


7 


/s 
















Group # 


Specimen Type 


^otoh (in) 


Dfin) 


On (MPa) 


ao 


F(a/R) 


9(a/R) 


g'(a/R) 


X=(g/g-)D 


Y (1/MPa) A 2 


1 


r/R=0.08, a07R=25 


0.75 


3 


9.632 


0.25 


1.1892 


0.1125 


0.4073 


0.01053 


0.003 


2 


r/R=0.08, a0/R=.45 


1.35 


3 


6.639 


0.45 


1.3146 


0.2475 


0.9999 


0.00943 


0.002 


3 


r/R=0.12, aO/R=25 


0.75 


3 


9.277 


0.25 


1.4364 


0.1642 


0.2491 


0.02511 


0.005 


4 


r/R=0.12, a0/R=.45 


1.35 


3 


7.057 


0.45 


1.4296 


0.2927 


1.1081 


0.01007 


0.002 


5 


r/R=0.08, aO/R=. 25 


1 


4 


9.267 


0.25 


1.1892 


0.1125 


0.4073 


0.01404 


0.003 


6 


r/R=0.08, a0/R=45 


1.8 


4 


7.598 


0.45 


1.3146 


0.2475 


0.9999 


0.01258 


0.002 


7 


r/R=0.12, aO/R=.25 


1 


4 


8.880 


0.25 


1.4364 


0.1642 


0.2491 


0.03348 


0.005 


8 


r/R=0.12, aO/R=.45 


1.8 


4 


7.328 


0.45 


1.4296 


0.2927 


1.1081 


0.01342 


0.002 




















Slope = 


0.1437 




















Intercept 3 


0.000571 






Ku (MPa-m A 0.5) = 


2.638 




















O (m)= 


0.0039743 



CN 
< 

O 
Q. 



>- 



0.006 -, 



0.004 - 



0.002 



0.000 



Experiment Data from all 8 Cylinder Groups 
Linear Regression for Fracture Parameters 



l — i — i — i — | — r 
0.005 0.010 



Coefficient of determination, R-squared = 0.843063 
1 I I — I — I — I — I — T 



] I I I I | I I I I — | — I — I — I — I — | 

0.015 0.020 0.025 0.030 0.035 

X. (m) 



Figure G-9. Size Effect Method for Strain Rate = 7 /s, using Groups 1-8 



LIST OF REFERENCES 

1. Ross, C. A., Tedesco, J. W., and Kuennen, S. T., "Effects of Strain Rate on Concrete 
Strength," ACI Materials Journal . Volume 92-1, 1995, pp. 37-47. 

2. Davis, J. R. (Senior Editor), "Dynamic Fracture Testing," Metals Handbook. 9 th 
Edition. Volume 8. Mechanical Testing . American Society for Metals, Metals Park, Ohio, 
1985, pp. 259-297. 

3. John, R. and Shah, S. P., "Fracture of Concrete Subjected to Impact Loading," 
Cement. Concrete, and Aggregates . Volume 8, Number 1, Summer 1986, pp. 24-31. 

4. Shah, S. P., "Rate Effects on Fracture of Concrete," ASCE Materials Engineering 
Congress . Denver, Colorado, 1990. 

5. Oh, B. H., "Fracture Behavior of Concrete under High Rates of Loading," Engineering 
Fracture Mechanics . Volume 35, Number 1/2/3, 1990, pp. 327-332. 

6. Duffy, J. and Shih, C. F., "Dynamic Fracture Toughness Measurement Methods for 
Brittle and Ductile Materials," Proceedings of the 7 th International Conference on 
Fracture: Advances in Fracture Research . 1989, pp. 633-642. 

7. Rittel, D. and Maigre, H., "An Investigation of Dynamic Crack Initiation in PMMLA," 
Mechanics of Materials . Volume 23, 1996, pp. 229-239. 

8. Bacon, C, Farm, J., and Lataillade, J. L., "Dynamic Fracture Toughness Determined 
from Load-point Displacement," Experimental Mechanics . September 1994, pp. 217-223. 

9. Nallathambi, P., and Karihaloo, B. L., "Determination of Specimen-Size Independent 
Fracture Toughness of Plain Concrete," Magazine of Concrete Research . Volume 38, 
Number 135, 1986, pp. 67-76. 

10. Swartz, S. E., and Refai, T., "Influence of Size Effects on Opening Mode Fracture 
Parameters for Precracked Concrete Beams in Bending," Fracture of Concrete and Rock, 
Springer- Verlag, New York, 1988, pp. 243-254. 

11. Jenq, Y. S., and Shah, S. P., "A Two Parameter Fracture Model for Concrete," 
Journal of Engineering Mechanics . Volume 111, Number 4, 1985, pp. 1227-1241. 



155 



156 



12. Bazant, Z. P., and Kazemi, M. T., "Determination of Fracture Energy, Process Zone 
Length and Brittleness Number from Size Effect, with Application to Rock and 
Concrete," International Journal of Fracture . Volume 44, 1990, pp. 111-131. 

13. Tang, T., Ouyang, C, and Shah, S. P., "A Simple Method for Determining Material 
Fracture Parameters from Peak Loads," ACI Materials Journal . Volume 93, Number 2, 
1996, pp. 147-157. 

14. Bazant, Z. P., "Fracture Mechanics and Strain- Softening in Concrete, in Proceedings 
of U.S. -Japan Seminar on Finite Element Analysis of Reinforced Concrete Structures," 
American Society of Civil Engineers, New York, 1986, pp. 121-150. 

15. RELEM Committee on Fracture Mechanics of Concrete - Test Methods, "Size-Effect 
Method for Determining Fracture Energy and Process Zone Size of Concrete," Materials 
and Structures . Volume 23, Number 138, November 1990, pp. 461-465. 

16. Ouyang, C, Tang, T., and Shah, S. P., "Relationship Between Fracture Parameters 
from Two Parameter Fracture Model and from Size Effect Model," Materials and 
Structures . Volume 29, Number 186, March 1996, pp. 79-86. 

17. Timoshenko, S. P., and Goodier, J. N, Theory of Elasticity . McGraw-Hill, Inc., Third 
Edition, New York, 1970. 

18. Yang, S., Tang, T., Zollinger, D. G., and Gurjar, A, "Splitting Tension Tests to 
Determine Concrete Fracture Parameters by Peak-Load Method," Advanced Cement 
Based Materials . Volume 5, 1997, pp. 18-28. 

19. Tang, T. ; Bazant, Z. P., Yang, S., and Zollinger, D., "Variable-Notch One-Size Test 
Method for Fracture Energy and Process Zone Length," Engineering Fracture Mechanics . 
Volume 55, Number 3, 1996, pp. 383-404. 

20. Kolsky, H., "Investigation of the Mechanical Properties of Materials at High Strain- 
Rates of Loadin g." Proceedings of the Physical Society . Section B, Volume 62, 1949, pp 
676-700. 

21. Nicholas, T., Impact Dynamics . J. Wiley and Sons, New York, 1982, pp. 277-332. 

22. Lindholm, U.S., "Some Experiments with the Split Hopkinson Presure Bar," Journ al 
of the Mechanics and Physics of Solids . Volume 12, 1964, pp. 317-335. 

23. Ross, C.A., "Split-Hopkinson Pressure Bar Tests," ESL-TR-88-22, Engineering and 
Services Laboratory, Air Force Engineering and Services Center, Tyndall Air Force Base 
Florida, March 1989. 



157 



24. John, R, and Shah, S. P., "Effect of High Strength and Rate of Loading of Fracture 
Parameters of Concrete," SEM/RILEM International Conference on Fracture of Concrete 
and Rock, Shah, S. P., and Swartz, S. E., Editors, Houston, Texas, June 1987, pp. 35-52. 

25. Gopalaratnam, V. S., and Shah, S. P., "Properties of Steel Fiber Reinforced Concrete 
Subjected to Impact Loading," ACI Structural Journal . Volume 83, Number 1, 1985, pp. 
117. 

26. Tinic, C, and Bruhwiler, E., "Effects of Compressive Loads on the Tensile Strength 
of Concrete at High Strain Rates," International Journal of Cement. Composites, and 
Lightweight Concrete . Volume 7, Number 2, 1985, pp. 103-108. 

27. Gorham, D. A. and Wu, X. J., "An Emprical Method of Dispersion Correction in the 
Compressive Hopkinson Bar Test," Journal de Physique. IV. Collogue.. Volume 7, 
Number 3, August 1997, p. 223. 

28. Kolksy, H, Stress Waves in Solids . Oxford University Press, London, 1953, pp. 54-73. 

29. Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity . 4 th Edition, 
Dover Publications, New York, 1927, pp. 427-428. 

30. Davies, R. M., Philosophical Transactions . Volume 240, 1948, p. 375. 

31. ADINA, A Finite Element Computer Program for Automatic Dynamic Incremental 
Nonlinear Analysis, ADINA R&D, Incorporated, Watertown, Massachusetts, September 
1990. 

32. Bathe, K. J., Finite Element Procedures in Engineering Analysis . Prentice-Hall, 
Incorporated, Englewood Cliffs, New Jersey, 1982. 

33. Irwin, G. R., "Onset of Fast Crack Propagation in High Strength Steel and Aluminum 
Alloys," Sagamore Research Conference Proceeding s, Volume 2, 1956, pp. 289-305. 

34. Ravichandran, G. and Chen, W., "Dynamic Failure of Brittle Material under Uniaxial 
Compression. " Experiments in Micromechanics of Failure Resistant Materials . Volume 
130, Editor K. S. Kim, ASME, New York, 1991, pp. 85-90. 



BIOGRAPHICAL SKETCH 

David Edward Lambert was born in August 1963 at Eglin Air Force Base, Florida, 
and attended primary and secondary school at the base and surrounding community. He 
entered Okaloosa-Walton Junior College in August 1981 and graduated with an 
Associates of Arts degree in engineering technology. He continued undergraduate studies 
at Florida State University from August 1984 until May 1987 when he graduated with a 
Bachelor of Science in Mechanical Engineering. During the FSU tenure, he participated in 
two semesters of cooperative education training with the Civil Engineering Department's 
Energy Management Office at Eglin Air Force Base. Following the undergraduate degree 
he was employed as a research engineer by the Air Force Armament Laboratory at Eglin 
Air Force Base in May 1987. 

David then entered the graduate school at the University of Florida's Graduate 
Engineering and Research Center at Eglin Air Force Base, Florida, in August 1987. After 
four years of part-time status, he graduated with a Master of Engineering in engineering 
mechanics in December 1991. 

He obtained a fellowship from the Air Force and reentered the graduate school at 
the University of Florida GERC in August 1995. He completed the requirements for the 
Doctor of Philosophy degree in December 1998. 

David is married to the former Deborah Diane Trippe. They have a son, Jordan 
Jayce, and a daughter, Christa RaeAnn. 



158 






I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy 



"' Allpn Rr»cc rhair J 



C. Allen Ross, Chair 
Professor Emeritus of Aerospace Engineering, 
Mechanics, and Engineering Science 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 





es E. Miltofi 
ngineer Emeritus of Aerospace Engineering, 
Mechanics, and Engineering Science 



I certify that 1 have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 



Edward K. Walsh 

Professor of Aerospace Engineering, 
Mechanics, and Engineering Science 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 










Davy M^Belk 

Assistant Professor of Aerospace Engineering, 

Mechanics, and Engineering Science 



I certify that I have read this study and that in my opinion it conforms to 
acceptable standards of scholarly presentation and is fully adequate, in scope and quality, 
as a dissertation for the degree of Doctor of Philosophy. 

cfi^t. a dl — 

Christopher S. Anderson 
Assistant Professor of Electrical and 
Computer Engineering 



This dissertation was submitted to the Graduate Faculty of the College of 
Engineering and to the Graduate School and was accepted as partial fulfillment of the 
requirements for the degree of Doctor of Philosophy. 

December 1998 r £-^_ 

/— Winfred M. Phillips 

Dean, College of Engineering 



M. Jack Ohanian 

Interim Dean, The Graduate School 



LD 

1780 
199 J 

■ LZ2Z 



UNIVERSITY OF FLORIDA 



3 1262 08555 1157